Ortiz did that for the second day in a row before Wednesday's game and will continue to do so throughout Boston's four-game series at Yankee Stadium.
If things go as Ortiz hopes, he will be taking live batting practice in the cage at some point next week, when the Red Sox return to Fenway for a six-game homestand that will close out the first half.
"I don't know, I'll probably be hitting in the field maybe when I get back to Boston," said Ortiz, who sustained a partial tear of the tendon sheath in his left wrist on May 31. "It all depends on how I feel. This is kind of a weird injury, and it's because one day you feel really good, and another day you feel kind of weak. It seems like you have to do what the trainers tell you to do and just hang in there."
Even in the elementary drills that Ortiz restarted on Tuesday, he felt a little like a fish out of water.
"I looked all [messed] up doing soft toss," he laughed. "I never thought I would have such a hard time hitting soft toss. I took like 25 swings, 25 soft tosses. At the end I started catching up. I'm going to go through the same thing today, and we'll see."
Ortiz is getting closer, however, and that can be seen in his demeanor, which has been noticeably more upbeat in recent days.
"Every day I feel better," he said. "Every day I discover something new, but in a good way. You know me, my problem might be because I swing so hard, you know, I've got to be 100 percent to play, because otherwise I could have a setback, and I don't want that. I don't think this team needs that. That's why I've got to take my time before I start playing. But I'm feeling good, though."
For the first time since Ortiz went down, the Red Sox have suffered a bit of a slump on this road trip, losing four in a row leading into Wednesday night's finale at Tropicana Field.
"Important players are always going to be important players," said Ortiz. "In our lineup, I make a difference when I'm playing. When I'm out of it, you can tell. I've been here for a while and doing my thing. When something like this happens, this team is going to feel it. You'll notice a difference if anyone else goes down. Everyday players, you always want to have them out there."
And that's even more true when you're talking about a game-changing bat like Ortiz's.
"He's doing great," said manager Terry Francona. "He's getting antsy, which is really good news. You see him, he's kind of smiling. He feels good about himself, that's good."
Ian Browne is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.