The next step would be a Minor League rehab assignment, though that's not something the Blue Jays are even discussing at the moment. For now, Toronto still wants Hill to take his time, even though he's itching to return to the lineup.
"He's dying to get back," Gaston said. "He really feels bad he's not here playing. But I told him that we'd like to have him back healthy, because that's something that you just don't mess around with, as far as concussions.
"We'd love to have him back, but we want him back when he's able to come. If he was ready, he'd come right now."
Hill, 26, was struck on the side of the head during a collision with Toronto shortstop David Eckstein on May 29 in Oakland. Since then, Hill has struggled with headaches, dizziness and sleepless nights. The Jays' medical staff has consulted numerous neurologists to help determine the extent of Hill's injury.
Hill had been taking part in light hitting and fielding drills earlier this month. That stopped after he was instructed to sit out for two weeks after visiting a concussion specialist at the University of Pittsburgh Medical Center during Toronto's trip to Pittsburgh on June 20-22.
"We don't know when he's going to be back," Gaston said. "Hopefully, he's going to be back soon, but we have no idea when he's going to be back."
Helping matters is the fact that the Blue Jays have received good play from fill-in second basemen Joe Inglett and Marco Scutaro.
While Hill has been sidelines, Inglett has hit .370 with two homers, 10 RBIs and a .452 on-base percentage in 23 games. Scutaro has managed only a .233 average during that same span, but he's provided good defense and has performed better lately at the top of Toronto's order.
"They've done well," Gaston said. "Joe's done a great job and Scutaro has played well there, too. We've missed [Hill], but these guys have been filling in really well for him."
Hill isn't the lone Toronto player rehabbing down in Florida. Blue Jays starter Shaun Marcum, who has been on the 15-day disabled list with a strained right elbow since June 19, began a throwing program on Saturday and is currently on target to return to the rotation following the All-Star break.
Outfielder Shannon Stewart -- out since June 8 with a severly sprained right ankle -- is still experiencing pain when running, but has been able to do light work off a tee and in batting practice. Reliever Jeremy Accardo (strained right forearm on May 10) is throwing, but hasn't returned to a mound yet since suffering a setback earlier this month.