The 22-year-old goalkeeper was named between the posts for Pools' defeat to Harrogate on Tuesday night.
Thompson became a free agent during the summer after leaving Newcastle United without making a senior appearance for the Premier League club.
He spent last season with Ayr United in the Scottish Championship, gaining further first-team experience during his time in Scotland.
The goalkeeper has also played in the English Football League, having previously spent time on loan with Chesterfield and Northampton Town.
Hartlepool have moved quickly to bring Thompson into their squad, with his arrival providing an immediate option in goal.
His inclusion against Harrogate means he will make his first appearance for the club having arrived on a deal initially covering only the short term.
Thompson's route into senior football has included loan spells rather than a first-team opportunity at Newcastle, where he developed before being released this summer.
The move to Hartlepool gives him another chance to establish himself in the professional game, while Pools will hope his experience with both EFL and Scottish Championship clubs can strengthen their squad.
His previous spells with Chesterfield and Northampton Town mean he already has experience of the demands of English league football despite still being only 22.
The immediate task is Harrogate, with Thompson handed his Hartlepool debut from the start.