The 29-year-old's seventh and final appearance with Triple-A Sacramento came on Tuesday, when he threw one scoreless inning.
"I couldn't sleep last night, for sure," O'Flaherty said Thursday. "It's cool to join a team this good and a bullpen this deep where there's not going to be too much pressure on me to shoulder too much of a load. I can just kind of get in where I fit in and help any way I can."
Manager Bob Melvin would ideally like to start by using O'Flaherty in a few low-leverage situations.
"I'd like to get him in some games before we get him pitching in the seventh or eighth inning with two on," Melvin said. "In a perfect world, you'd like to get him in a couple of settings where it's a bit of a soft landing, but we'll see if that transpires."
Over the past five seasons, O'Flaherty has been one of the premier left-handed relievers in baseball, compiling a 1.99 ERA in 249 1/3 frames with Atlanta. Lefties have hit .200 against him in his career.
"His four- and two-seam [fastballs], he gets a lot of late movement on it," said Melvin. "He's got a really good slider that he can throw down and in to a righty, away from a lefty. He can do it all."
O'Flaherty has yet to pitch on consecutive days, but he believes that step will come shortly. His rehab process went smoothly, with no major setbacks, and he said his velocity is similar to before the surgery.
"Maybe not topping out the same, but I usually don't get my velocity until about this time of year anyway," he said. "Getting a late start, hopefully there's a little more in the tank, but I can definitely pitch where I'm at."
Francis was claimed off waivers by Oakland on May 18 and went 0-1 with a 6.08 ERA in nine relief appearances. He picked up his first career save on Saturday.
Aaron Leibowitz is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.