Coronavirus tests might be available for people without symptoms after new laboratories open in November, Wales' first minister has said.
Mark Drakeford's suggestion follows reports of people failing to get tests or being told to travel long distances.
He also defended a decision to use the UK-wide system of Lighthouse labs which have been blamed for the problems.
Some tests are processed by those labs, while others are sent to labs run by Public Health Wales (PHW).
Facing questions in the Senedd, Mr Drakeford said PHW laboratories in Cardiff, Swansea and Rhyl will start processing swabs 24 hours a day next month.
Staff are also being recruited to work in "hot labs" at hospitals, due to open in November.
Job interviews started on Monday, with 3,000 people applying for 160 posts, Mr Drakeford said.
The Welsh Government announced an extra £32m to speed up testing last month.
At question time, Plaid Cymru leader Adam Price asked whether the labs could open sooner given the "growing urgency" of problems in the testing regime.
When the extra lab space is available, he asked whether the Welsh Government would consider extending tests to people who do not have symptoms but who are identified through the contact-tracing system.
At the moment - and with the system under strain - people are being told only to get a test if they have symptoms of Covid-19.
Mr Drakeford said the sooner staff could be appointed to the new labs, the sooner they will be active.
He added: "When we have more capacity in that way we will be able to think again about who we test, when we test them including - I'm not suggesting we've made that decision at all - but it will allow us to consider the issue of asymptomatic testing in a different way."
He rejected claims by Plaid Senedd member Rhun ap Iorwerth that the Welsh Government was wrong to "put its faith" in the UK-wide system.
The first minister said the lighthouse labs had been working well until three weeks ago.
"The challenge now is to do the work on a UK level to put the lighthouse labs back where they were in August and that's what we want to support."