Thompson allowed one hit, allowing two inherited runners to score, over a third of an inning in Thursday's 5-4 win over the Reds.
Thompson (0-1) was tagged with Monday's loss to the Cardinals, allowing one run on one hit while striking out one over one-plus innings.
Thompson allowed a hit and struck out one over 1.1 scoreless innings in Friday's 6-5 extra-inning loss to Atlanta.
Thompson moves up a spot in the bullpen following the news that closer Paul Sewald (oblique) will open the season on the injured list.
Thompson allowed one hit while striking out two over a scoreless inning in Wednesday's 3-0 win over the White Sox.
The Diamondbacks selected Thompson's contract from Triple-A Reno on Sunday.
Thompson (undisclosed) signed a minor-league contract with the Diamondbacks on Sunday, Alex Weiner of ArizonaSports.com
The Rays released Thompson on Wednesday.
The Rays designated Thompson (undisclosed) for assignment Wednesday, Tricia Whitaker of Bally Sports Sun reports.
Thompson was optioned to Triple-A Durham on Wednesday, Marc Topkin of the Tampa Bay Times reports.