Catering giant Compass Group and Whitbread, one of Britain's largest hotel chains, have found horse DNA in products sold as beef, it has emerged.
Horsemeat has also been found in cottage pies supplied to 47 schools in Lancashire - they have been withdrawn.
But the Food Standards Agency says that after 2,501 fresh tests no new products have been identified as containing more than 1% horsemeat.
It said the 29 positive results were on seven previously withdrawn products.
FSA chief executive Catherine Brown said she remained "confident" that the testing was the right way to address the issue.
"It is industry's responsibility to get this right - not the government's - and we consider that a comprehensive testing programme at all points of the supply chain and in all sectors is an essential step in addressing this issue.
"And as this programme of testing and publishing results continues, and as action is taken to tackle this issue in supply chains across Europe, we will reach the point where we can say with confidence that horse meat is no longer illegally entering the UK food chain."
'Shocked and dismayed'
Compass Group, one of the biggest school food providers in the UK, says its tests have found between 5% and 30% horse DNA in burgers it sold in Ireland and Northern Ireland.
It says the burgers were supplied by Rangeland Foods in County Monaghan, which previously withdrew more than 9,000 burgers made for the UK market after some were found to contain horsemeat.
Compass said the Rangeland burgers had been supplied to 13 sites in the Irish Republic and 27 in Northern Ireland, mostly offices but including two unnamed secondary schools.
It described the situation as "totally unacceptable" and said all affected sites had been given "unreserved apologies".
And Whitbread, which owns Premier Inn, Beefeater Grill and Brewers Fayre, says it found horse DNA in two products - burgers supplied by food supplier Paragon Quality Foods and lasagne from Brakes Brothers.
Creative Foods, owned by Brakes Brothers, and which caters for schools and hospitals in the UK, had used frozen beef mince for lasagne from Hampshire-based supplier Pinnacle Foods, some of which has now been found to contain horse DNA.
Creative says it has now stopped using Pinnacle Foods as a supplier, and Brakes is also recalling one of its own-brand lasagne products as a precaution.
Pinnacle's managing director Graham Reed said this development was "a complete surprise and shock to us".
He added: "We are devastated by the news, and working very hard to trace back where the offending material may have come from.
"We have never ever knowingly had equine material on our premises or indeed ever dealt in horsemeat. We will be working very closely with FSA and customers alike to get to the bottom of it."
Environment Secretary Owen Paterson told the BBC he expected the food industry to have completed their horsemeat tests by the end of next week.
He said it was for the industry to "get out there and reassure the public".
Asked about the retailers' response to the crisis, he said that was "for the food businesses themselves to decide, they are responsible for the quality and integrity of what they present to the public", before adding that it was not for him to "micro-manage" food businesses.
But Labour's Mary Creagh said the public would be "shocked and dismayed that horsemeat has now been found in schools and hospitals" and called for the prime minister to order the FSA to speed up its testing.
In other developments:
- The European Union has decided immediately to start testing for the presence of unlabelled horsemeat in foods across the Continent. Tests will also be carried out for the presence of residues of the drug phenylbutazone, used on horses but banned from entering the food chain
- The FSA says that following three arrests made on Thursday in Wales and West Yorkshire in relation to suspected fraud regarding the horsemeat scandal, there have now been seizures of evidence at Flexi Foods Ltd in Hull and at Dinos & Sons Continental Foods Ltd in Tottenham, London
- Flexi Foods, based at Hull's Newland Science Park, confirmed it was "co-operating" with the FSA while Dinos & Sons said it had been "asked to clarify its position in respect of the transportation and storage of frozen beef that was imported by, and belonged to, a third party that the FSA is investigating"
- A company which provides school meals to 128 schools in Bristol - Eden Foodservice - has withdrawn all of its processed beef products from its menus as a precautionary measure
- On Friday, it emerged beefburgers containing horsemeat had been withdrawn from hospitals in Northern Ireland
- French meat processing company Spanghero apologised to British consumers but denied knowingly selling horsemeat labelled as beef
A group of 11 food suppliers, including Tesco and Asda, issued a letter on Friday stating they shared shoppers' "anger and outrage" and rejecting government criticism they "remained silent" over the crisis.
The letter was signed by chief executive of Tesco, Philip Clarke, Asda Stores boss, Andy Clarke, the chief executive of J Sainsbury, Justin King, and Dalton Philips, chief executive of Wm Morrison Supermarkets, among others. Several also released the results of their tests.
Iceland said that all of its own brand beef products have been found to be free from horsemeat.
The Co-Operative Group said 59 of its 102 own-brand minced beef products have been tested so far, with all found to be clear of horsemeat. And Morrisons says 68 test results on its products have not found horsemeat, with more results still to come.
Tesco says tests on 149 of its products are clear, and Sainsbury's say their tests show no horse adulteration. Waitrose says it has conducted about 40 tests, none of which showed the presence of horsemeat.
Last month, Irish food inspectors said they had found horsemeat in beefburgers made by firms in the Irish Republic and the UK, and sold by a number of UK supermarket chains, including Tesco, Iceland, Aldi and Lidl.
Since then, a growing number of UK retailers have recalled processed beef products found to contain horse DNA.
Some shops have already recalled products found to be adulterated, including Asda, which withdrew a beef Bolognese sauce on Thursday - the first fresh beef product to be involved.
Aldi, Tesco and Findus have also withdrawn some beef-based ready meals.
French food producer makes order
Comigel HQ in Metz, north-east France, asks its subsidiary, Tavola in Luxembourg, to make food products - including beef lasagne for Findus.
Factory orders meat
The Tavola factory orders the meat from Spanghero in the south of France.
Spanghero contacts a subcontractor in Cyprus to source the meat.
Subcontractor enlists trader
The Cypriot subcontractor in turn contacts a trader in the Netherlands.
Trader orders from Romania
The trader in the Netherlands places an order for meat with abattoirs in Romania.
Abattoirs send meat to France
The meat from the abattoirs travels to Spanghero in France. However, Romania rejects claims that it was responsible for wrongly describing the horsemeat from its abattoirs as beef. Horsemeat is always labelled as such, they say. The Romanian authorities claim records show orders had been for horse carcass - easily distinguishable from beef.
Meat used to make products
Spanghero sends the meat to the Comigel subsidiary’s factory in Luxembourg before the finished products are supplied to Findus and retailers across Europe, including the UK. The president of Comigel says the company was unaware the meat was coming from abroad.
Horsemeat found in Ireland and UK
Tests by Irish authorities have found equine DNA in beefburgers made by firms in the Irish Republic and the UK. Traces of horsemeat have also been found in stored meat at another plant in Ireland and one in Northern Ireland. In mainland Britain, police and officials probing alleged horsemeat mislabelling have carried out raids at a slaughterhouse in West Yorkshire and a meat firm near Aberystwyth. Three men were later arrested on suspicion of offences under the Fraud Act..