Prioritising coronavirus testing in high-risk areas has led to shortages in other places, leading to some people with symptoms being asked to drive more than 100 miles for a swab.
The government says areas with fewer Covid-19 cases have had their testing capacity reduced to cope with outbreaks elsewhere.
But public health experts warn this could miss the start of new spikes.
Meanwhile, a £500m fund to trial a 20-minute saliva test has been unveiled.
Health Secretary Matt Hancock told the BBC that testing was being focused on areas with outbreaks where case numbers are much higher, but the "vast majority" of people could get a test "close to home".
He admitted there were "operational challenges", adding that the trial of new rapid tests could "solve the problem".
Groups of people including staff and students at the University of Southampton and four schools in the Hampshire city will take part in the trial.
There will also be a trial in Salford of the benefits of repeatedly testing a population.
"Short of a vaccine, this is the best chance we have of reducing social distancing whilst controlling the virus, especially with winter coming and all the challenges that brings," the health secretary told BBC Radio 4's Today programme.
'I was told to travel 250 miles for a test'
When David Llewellyn tried to book a test on Sunday he was directed to a test centre near Blackburn - 250 miles from his home in Stowmarket, Suffolk.
"I thought I'd entered my home location incorrectly so went through the booking process again, only to get the same location for testing.
"Needless to say I have subsequently ordered a home testing kit, which despite three automated email reminders to take the test has yet to arrive. I'd do the test if I had it.
"I think it's ridiculous. The idea that the test centres near me in rural Suffolk are at capacity is unlikely and whilst I accept that Covid is unpredictable and the science is changing all the time, I don't think the government have got their act together.
"The fact I've been directed 250 miles away in order to get a test shows they haven't got it right yet."
It came as a postcode search of the government's coronavirus test booking service revealed:
People with symptoms living in London are being directed between 50 and 135 miles away, including to testing sites in Cardiff and the Isle of Wight A postcode in Devon was directed to a testing centre 109 miles away in Carmarthen One in Worthing was directed 40 miles away One in Cumbria was sent 50 miles away to Dumfries In Sheffield on Wednesday morning the closest available testing centre was 20 miles awayThese distances are the ones that are given on the government website, but they appear to have been calculated as the crow flies, rather than being a true reflection of how long it would take to drive - for example, the Devon to Carmarthen route was actually a 206-mile drive according to Google Maps.
Dr Zoe Norris, a GP in Yorkshire, told BBC Breakfast such journeys would be a "huge undertaking" for those with coronavirus symptoms, which include a fever and persistent cough.
"I've certainly got patients who would not manage to drive those distances," she added.
Rationing
National testing capacity has not reduced, but it's already being outstripped by rising applications for tests.
The Department of Health and Social Care (DHSC) has responded by rationing testing slots, with fewer tests available in areas with fewer cases.
Dr Stephen Baker at the University of Cambridge, who has been conducting testing for the area's hospital staff, said this was a "pragmatic move from the government to focus resources where they are most needed", but the strategy could become a problem if more cases developed outside the current areas of concern.
People with symptoms and essential workers can still apply for home kits, although these take longer to return a result.
It has also been taking longer for people who do go to a drive-through testing centre to receive their results.
NHS Test and Trace figures show in the week of 13-19 August, 41% of in-person tests at mobile testing units and drive-through centres gave results within 24 hours. That's down from 61% the week before.
For all routes combined, including home kits, 20% of tests had results back within 24 hours, compared with 29% the week before.
Is demand for tests outstripping supply?
Regular mass testing of people without symptoms is seen by ministers as the best passport to further easing of lockdown restrictions until a vaccine is developed.
Simple kits with rapid results, used weekly, can give assurances people don't have coronavirus. That's what the new funding for trials and pilot schemes is all about - but that's a long-term goal.
The reality right now is capacity shortages and long-distance drives for some people needing tests.
Demand seems to be outstripping supply even after a big expansion in testing sites and laboratories.
Public health experts have warned these issues need to be urgently addressed to ensure that local outbreaks can be properly monitored and tackled as winter approaches.
'High demand'
Paul Hunter, a public health expert and professor of medicine at the University of East Anglia, said these issues could act as "big disincentive to being tested" and result in missing local increases "early enough to maybe stop more widespread infection".
A DHSC spokesperson said: "There is a high demand for tests and our laboratories continue to turn test results around as quickly as possible.
"To make sure we stay in control of this virus we are targeting our testing capacity at the areas that need it most, including those where there is an outbreak, as well as prioritising at-risk groups."
But virologist Prof Nicola Stonehouse at the University of Leeds said: "Although cases are low we need to remember that the pandemic started from a single case and that only a small number of people entering the UK (mainly from Italy) resulted in the large number of cases earlier this year."
"The key to controlling the pandemic has to be through testing. And this needs to be made as accessible as possible," she added.
Image copyright Getty Images Image caption The government said there is "high demand" for coronavirus testsThe government has pledged to increase its capacity to 500,000 tests a day by the end of October.
Currently, it says, testing capacity is about 350,000 a day - but only just over half of that is being used.
Daily testing has only broken 200,000 on one day, despite the government hailing reaching its target of having that level of capacity at the end of May.
The DHSC stressed that booking slots were added in the evening for morning appointments and in the morning for afternoon appointments, so more local slots might become available through the day.
The website states: "This service is currently very busy. If you are unable to book a test now or the location and time is not convenient for you, please try again in a few hours when more tests should be available.
"If no tests are available online, do not call helplines to get a test. You will not be able to get a test through the helplines."
Follow Rachel on Twitter
Have you been asked to travel long distances to get a test? Please share your experiences by emailing haveyoursay@bbc.co.uk.
Please include a contact number if you are willing to speak to a BBC journalist. You can also get in touch in the following ways:
WhatsApp: +44 7756 165803 Tweet: @BBC_HaveYourSay Please read our terms & conditions and privacy policy