Hudson earned a hold against the Giants on Thursday, walking one batter and striking out two in a scoreless sixth inning.
Hudson did not walk or strike out a batter during a two-out save in Sunday's 9-6 win over the Red Sox.
Hudson wasn't available out of the bullpen for Sunday's 4-3 loss to the Tigers due to back stiffness, Fabian Ardaya of The Athletic reports.
Hudson earned a hold against Kansas City on Sunday, striking out two batters in a perfect eighth inning.
Hudson (2-1) gave up two hits and a walk while striking out one in a scoreless ninth inning during the first game of Tuesday's doubleheader against the Mets, recording the win in a 5-2 extra-inning victory.
Hudson is the likely favorite for save opportunities after Dodgers closer Evan Phillips (hamstring) was placed on the 15-day injured list Sunday.
The Dodgers selected Hudson's contract from Triple-A Oklahoma City on Saturday, J.P. Hoornstra of DodgersNation.com reports.
Dodgers manager Dave Roberts said Wednesday that Hudson "will be in our bullpen," Mike DiGiovanna of the Los Angeles Times reports.
Dodgers manager Dave Robert said Friday that Hudson (knee) is trending toward being on the active roster for the March 20-21 series against the Padres in South Korea, Fabian Ardaya of The Athletic reports.
The Dodgers re-signed Hudson (knee) to a minor-league contract Wednesday, Jeff Passan of ESPN.com reports.