Doolittle made 26 pitches, mixing in all of his pitches on Thursday.
"That's the best bullpen I've seen him throw," manager Bob Melvin said, per MLB.com. "I think he feels psychologically a little bit better about letting it go. It seemed to me that he was guarding it a little bit more so the last time. He definitely was not this time. It seemed like there was a lot more whip."
The left-hander, who is out with a strained left shoulder, is expected to have another bullpen session on Saturday. If all goes well, he could go out on a rehab assignment shortly thereafter.
Bassitt went seven innings, scattering six hits while striking out six and walking just one in what was the best of his five starts this season. However, Bassitt's record dropped to 0-4.
He allowed one of the runs to score on an RBI groundout, but was able to escape the inning without any further damage. He came back out and worked a scoreless ninth inning to preserve the victory.
Wilson was tasked with getting the save on the day that closer Joakim Soria was dealt to the Pirates. It is not known if Wilson will be inheriting the closing duties, or if manager Brad Ausmus will be using more of a committee approach.
Wright fell behind early, giving up a two-run, first-inning home run, but stabilized himself and went six innings, scattering six hits. Wright's knuckleball fluttered effectively, as he struck out eight Chicago hitters.
With the win, Wright (4-4) picked up his first victory since being enlisted to make spot starts in the Red Sox's rotation.
It only took Osuna six pitches to earn his sixth save of the season -- putting him in the team lead for that category. Osuna registered three ground-ball outs to retire the side. Osuna now has four saves in July.
To make room, the team optioned Tommy Layne to Pawtucket. Layne was 0-1, 3.51 with a save for Boston.
Machi struggled with the Giants this season, posting a 5.14 ERA over 35 innings. But manager John Farrell explains the cause of Machi's struggles.
"His fastball velocity has been consistently between 92 to 94 mph. His damage has come with his slider," Farrell said, per MLB.com. "It hasn't been as sharp as it was the last two years. That's the pitch that we have to hone in on and regain that consistency.
"There is a little bit of a challenge bringing him in here with our situation with a need for innings right away with his inactivity. We'll see what he has."
Machi had made just one appearance for the Giants since the All-Star break.
To make room on the 40-man and 25-man rosters, Daniel Nava has been designated for assignment.
The trade comes just over a year after Soria was dealt to the Tigers by the Rangers last season. This year, Soria has recorded 23 saves in 26 chances for the Tigers. He has 36 strikeouts in 41 innings pitched. He will join Mark Melancon as a formidable 1-2 punch at the back end of the Pirates' bullpen.
The Tigers, who have officially announced the trade, will receive minor-league infielder JaCoby Jones from the Pirates. Jones was the Pirates' No. 12 prospect, and was batting .260 with 10 home runs and 60 RBI between Class A Bradenton and Double-A Altoona. He was a third-round pick by the Pirates in 2013.
Rasmus is 0-1 with a save and a 2.35 ERA at Salt Lake City. He last pitched for the Angels last season.
Rasmus takes the spot of Mike Morin, who was optioned after Wednesday's game.
Mitchell pitched in six games for the Yankees in June and July, compiling a 2.89 ERA and fanning nine in 9 1/3 innings. Goody, who has yet to pitch in the big leagues, has been tearing it up in the minors. He has struck out 70 in 48 1/3 innings and owns a 1.68 ERA.
Cotham earned a spot start for the Yankees on Wednesday and performed well before his removal in the second inning. He gave up two hits and fanned four.