Up to five extra mobile testing units will be deployed to Covid-19 hotspots in Wales this week, Wales' Health Minister Vaughan Gething has said.
He said extra lanes at drive-through centres and tests processed at a Welsh laboratory were also being considered.
People trying to get tests have spoken of their frustration over the length of time to book and distance to travel.
GP and Plaid member of the Senedd (MS) Dr Dai Lloyd has called on Mr Gething to "get a grip" on the issue.
He told Claire Summers on BBC Radio Wales the health minister had "opted into this UK system" so "can't say 'it's not my fault, it is all down to UK'."
The large private sector Lighthouse Laboratories which analyse test swabs from all the UK nations have been under strain to process them all.
But Mr Gething said that Wales would not stop using Lighthouse Laboratories: "I think dynamiting the system and thinking we can start again within a few days... I don't think that would be sensible at all... we don't have the additional money."
He said issues with testing were "a point of real frustration for people across Wales and indeed other parts of the UK", adding: "We're already looking at what we can do with our own testing resources here in Wales".
He told Radio Wales he and the Scottish health minister had jointly written a letter to UK Health Secretary Matt Hancock asking for an urgent discussion about how quickly issues would be resolved.
Caerphilly has been the second-hardest-hit part of the UK for infections in the last week, after Bolton, Greater Manchester, which tightened its local restrictions.
Analysis of recent case rates by the BBC also shows Rhondda Cynon Taff ranked 22nd and Merthyr Tydfil in 29th place.
What's been going wrong?
The UK government's testing system - part of its test, track and trace operation which Prime Minister Boris Johnson promised would be "world-beating" - has faced criticism in recent weeks.
A surge in demand for coronavirus tests has led to local shortages, with many people reporting problems with getting online bookings and being directed to test sites hundreds of miles from their homes.
On Monday, one testing site in Abercynon, Rhondda Cynon Taff, reportedly ran out of kits.
An official at the site told BBC Wales it had closed for a few hours while they fetched more tests from Swansea.
'It's someone else's fault'
Dr Lloyd, who is chair of the Senedd's health committee, said he was "extremely concerned because we seem to seeing a resurgence now of the virus".
"We're running out of testing kits in a situation that is run by private companies that have no link to the NHS," he said.
"We've fragmented the system now and there's lots of people involved."
He added: "It's a very complicated new private system overlaid on top of our usual NHS provision…
"Its impossible as opposition politicians to scrutinise exactly what's going on because everyone says it's someone else's fault."
How many tests are being carried out in Wales - and how quickly are results coming back?
- Latest figures out on Wednesday show 58,860 tests were processed in the week beginning 7 September
- Around 70% of these tests are processed by the UK network of Lighthouse labs, the rest are carried out by NHS Wales-run labs
- Friday 11 September saw a record number of Covid-19 test results authorised in a single day in Wales - 10,949
- But details of the turnaround times from Lighthouse laboratories for Welsh tests are not available for a second week running due to "data mapping issues"
- Before then, backlogs due to high demand and capacity issues were blamed for only 8.4% of home tests being processed by Lighthouse labs within a day and only 24% being processed within two days. This was for the week ending 24 August
- In the latest week, 74% of community and mass testing "in person" tests - including at mobile units - came back within a day from NHS Wales laboratories, according to the Welsh Government figures. This is the best performance on turnaround times since the data was first published in June
- 89% of hospital tests were processed in NHS Wales laboratories within a day. Altogether this involves nearly 16,000 tests handled by NHS Wales labs over the week
- The figures also show the testing in the Aneurin Bevan health board - where there has been a local lockdown in Caerphilly and a rise in infections in Newport. Last Wednesday alone, 2,374 people were tested - 1,242 in Caerphilly
- Across Wales in the latest week, 1,861 care home residents were tested and only seven people were positive.
A Welsh Government spokeswoman said: "The problems people have been experiencing when trying to book a test are directly related to the ongoing UK-wide issues with the Lighthouse Lab system, which we urgently need to see resolved at a UK level.
"The health minister [in Wales] has repeatedly raised this with the [UK government's] secretary of state for health.
"We are moving as much NHS Wales testing capacity as we can into areas where testing is needed the most, as well as taking urgent action to switch over testing facilities to Welsh laboratories to further increase capacity while the UK government resolves these issues with the Lighthouse Lab system."
Analysis from Paul Martin, BBC Wales political reporter
This is clearly not a problem that's going to solved immediately.
We have the two distinct amounts of capacity. There are the UK wide Lighthouse Labs which are causing problems all over the UK and which the Welsh Government is heavily invested in.
Then we have the NHS Wales labs that the Welsh Government is now trying to ramp up and switch over capacity in to.
If we go back to June, the Welsh Government was processing most of its tests in its NHS Wales labs. Then it moved over to the Lighthouse Labs because they were performing so well, turning things around quickly, and that looked like a good option.
But as demand has gone up across the UK, these problems have come to the fore and now we are in the situation where the Welsh Government is trying to reverse things, move capacity back over to its own set up.
But as Vaughan Gething said, it's not as simple as flicking a switch.
He wants to move more mobile units over and put more lanes on in the drive-through centres where the samples would then be kept in Wales and not sent to Lighthouse Labs.
The question now is how quickly and effectively he can do it.