Drew batted sixth vs. the Rangers on Monday, with Mike Lowell hitting seventh.
The Red Sox entered the day with a leadoff on-base percentage of .306, which ranked 27th out of the 30 teams in the Majors.
"It's something I'm sure we'll improve on," said Ellsbury. "From an offensive standpoint, we haven't hit our full potential yet."
While losing two out of three in Toronto over the weekend, the Red Sox hit .188.
The Red Sox are third in the American League in on-base percentage, fourth in slugging and fourth in runs. But most would agree that pitching is the reason they started the week leading the Yankees by one game in the AL East.
Are the bats due to bust out?
"I certainly hope so," Francona said. "But I don't know that you sit around and just wait for that. I think you just do what you're supposed to do and then you'll look up and somebody will say, 'You guys are on one of those runs.' You start talking about it or waiting for it, and you kind of get ahead of yourself."
Ellsbury certainly has the speed (40 stolen bases) any team would love out of a leadoff hitter. Initially moved out of the top spot because his on-base percentage was a little below what the Sox were looking for, he started getting on base more in the lower portion of the order.
"For me, they kind of wanted me to do the same thing at the bottom of the order -- just have quality ABs," Ellsbury said. "Leading off, I'm going to try to maybe see some pitches for the guys behind me. Then again, if he gives me pitches to hit, they want me to stay aggressive and put the ball in play."
Francona wasn't looking for any miracles from Ellsbury and the new-look lineup -- just some more consistent production.
"Let Jacoby hit there, keep everybody else in kind of some semblance of order," Francona said. "The one thing it does do is it kind of gives us that length in the lineup. We've got Lowell, [Jason Varitek] and [Jed] Lowrie seven, eight, nine. What I want [Ellsbury] to do is just do what he's been doing when he's hit down in the order. It seems like everybody we put in that leadoff spot, that's when they decide to quit getting on base. It's certainly not by design. So we'll give it a shot."