Tomko acts as if the job is his

Dodgers Manager Grady Little said he wants his starting pitchers to begin working on their hitting as opening day draws near. So perhaps there was some significance to the fact right-hander Brett Tomko had a new two-tone bat stuffed in his locker after Thursday’s 2-1 loss to the Florida Marlins.

Relievers, after all, have no need for a bat.

“They told me to get ready to be a starter and train in the off-season like I was going to start,” said Tomko, who pitched primarily out of the bullpen last season. “That’s what I did. So as far as I’m concerned that’s my job to win.”

Tomko is one of a half-dozen pitchers competing for the final spot in the Dodgers’ rotation, and he improved his chances with two dominant innings Thursday. It was his second scoreless outing in as many tries this spring.

No one else in the competition can make that claim after Joe Mays, who followed Tomko to the mound, gave up two runs to the first four Arizona batters he faced.

But Tomko, whose 231 career starts is second only to Jason Schmidt among Dodgers pitchers, said he can’t worry about what others are doing.

“The only thing I’m worried about is making pitches. The rest will take care of itself,” he said. “We’ll know in about 24 days.”

Betemit not concerned

With the third base job his to lose, Wilson Betemit has been able to relax this spring -- even after going hitless and striking out five times in his first 12 Grapefruit League at-bats.

“I’m not frustrated, because I know what I can do,” said Betemit, who hit .241 in 55 games with the Dodgers last season after coming over from Atlanta in a trade-deadline deal. “I know I can hit. Swing the bat, that’s all I have to do. Go to home plate and be aggressive.”

Apparently Betemit can be too aggressive, though, so hitting coach Eddie Murray has been working with him to stay back on the ball, especially with his hands.

That paid off Wednesday when Betemit picked up an RBI single for his first hit of the spring. Betemit joked afterward that coach Mariano Duncan asked the umpire for the ball as a memento of his first spring training hit as a Dodger.

Little says there will be many more to follow.

“It’s just a matter of time before he gets it going,” Little said. “And that hit might be just what he needs.”

Betemit lined out as a pinch-hitter in his only at-bat Thursday.

Slow starts

Betemit isn’t the only projected starter who has gotten off to a slow start this spring. Nomar Garciaparra was one for 11 before hitting a home run for the Dodgers’ only run Thursday. And catcher Russell Martin and left fielder Luis Gonzalez have combined for just three singles in 20 at-bats.

“I’m not worried,” Garciaparra said. “It’s about working on stuff, it’s not about the results.”

Memorial service

More than 200 people crowded into St. Helen Catholic Church for a memorial service for former Dodgers pitcher Clem Labine, who died last week at the age of 80. Hall of Famers Tom Lasorda and Sandy Koufax, former team owner Peter O’Malley and major league umpire Bruce Froemming were among those in attendance.... Kevin Hallinan, Major League Baseball’s vice president of security, will be in Dodgertown this morning as part of his regular spring tour briefing players on security issues.... Little said he may push Rafael Furcal’s return back to late next week. Furcal, who played catch and swung a bat Thursday, hasn’t played since the Dodgers’ March 1 exhibition opener because of a sore right shoulder.... Left-hander Joe Beimel, who complained of elbow pain his last time out, said he felt fine after a perfect inning of relief Thursday.

kevin.baxter@latimes.com