But the 27-year-old rookie is still in the role and he converted his first six-out save to help the bullpen hold on in a 7-5 victory over the Minnesota Twins on Tuesday night.
A giant banner with Hoffman's picture on it still says 596 after the 42-year-old right-hander got his last save on May 7, while Axford has saved all five of his chances since.
"I still don't consider myself a closer by any means," Axford said. "Maybe when I'm 591 more saves away from where I am right now, you can talk to me about it then and see where I'm at. ... I'll go in and do any job they want me to."
Casey McGehee hit a three-run homer and Rickie Weeks added a solo shot for Milwaukee, which built a 7-3 lead until reliever Todd Coffey allowed a two-run double to Delmon Young with no outs in the eighth.
"Sooner or later, Trevor's going to get a chance to close some games, but I don't think that's a question for tonight," Brewers manager Ken Macha said.
Axford nearly entered the game in the seventh because Zach Braddock was not with team because of a death in the family. But reliever Kameron Loe got Mauer to pop up and Axford's appearance was delayed an inning. Macha said Hoffman would've gotten the opportunity to close the game if Axford was used in that situation.
Brewers starter Chris Narveson (6-4) trailed 3-0 after failing to retire the first six batters. He didn't get an out until his 30th pitch, but helped himself and the Brewers' rally with the go-ahead RBI in the fourth after McGehee's homer and went on to retire 14 of the final 16 Twins he faced.
Twins starter Scott Baker (6-6) struck out seven coming off a career-best 12 in his last start. He allowed nine hits and five runs, dooming him from pitching past the sixth inning.
With this series, Milwaukee (30-40) is in a stretch of 16 of the next 20 games at home and needs to make up ground before the All-Star game.
"As far as critical, when you're this far behind, I guess you can put a label like that, but I don't like [it]," Macha said. "There's still a lot of baseball to be played."
The AL Central-leading Twins have been one of the Brewers' toughest tests in interleague play, winning the last five series between the clubs. Milwaukee got off to a good start to stop that streak.
"It's always a work in progress," said McGehee, who is hitting .194 in June. "I feel a lot better at the plate."
Jim Edmonds followed McGehee's homer with a double, just eluding the tag of the shortstop Punto and Narveson singled on a soft flare to left with two outs to give Milwaukee a 4-3 lead.
"You give up a four spot in an inning, you don't have a lot of room for error," Baker said.
Milwaukee tacked on three more runs after that, but needed them all when the Twins brought the tying run to the plate in the eighth and ninth.
Narveson fell behind 3-0 after loading the bases with no outs on a hit batter and two walks before allowing three straight singles, including a two-run effort by Morneau. He's allowed 15 runs in the first inning of his 11 starts, but settled down again. Minnesota got just two more hits off him through five innings.
"Sooner or later, he's got to realize he's got good stuff," Macha said. "That's a pretty good lineup and he throws four zeros after six straight hitters get on."
Kubel's 12-game hitting streak ended after entering as a pinch-hitter in the seventh. ... Brewers' fourth-round draft pick Hunter Morris was at Miller Park. He said he didn't have any plans for his $218,700 signing bonus. ... Macha stopped his briefing with the media when former Twins RHP and current TV commentator Bert Blyleven walked by. "If you haven't voted for him, what's wrong with you? Sixty shutouts! You've got to be kidding me. That should put him in the Hall of Fame alone."
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