The bodies of backpackers Hannah Witheridge, 23, from Norfolk, and David Miller, 24, from Jersey, were found on a beach on Koh Tao in September 2014.
In 2015 Burmese bar workers Zaw Lin and Wai Phyo, also known as Win Zaw Tun, were found guilty of the killings.
Their lawyers have until 23 May to file an appeal with the Supreme Court.
A Thai appeal court upheld the death sentences given to the two men in March.
Lawyers had been given 30 days to appeal to the Supreme Court, Thailand's final court of appeal.
The men claim DNA evidence used to secure their conviction was unsound and a forensic investigator previously testified that evidence had been improperly collected.
Lawyers for the convicted men have said they can still challenge the verdict, but because the defence team claims it is working on such limited resources they need more time to prepare a large amount of technical documents.
Miss Witheridge, a University of Essex student from Hemsby near Great Yarmouth, and Mr Miller, a civil and structural engineering graduate from the University of Leeds, were found on a beach bludgeoned to death.
A post-mortem examination showed Miss Witheridge had been raped.
Two men who face the death penalty in Thailand for murdering two British tourists have had their appeal deadline extended.