My kids are little so I never had the height problem, but my little boy was a climber from early on. Before we got him the "big boy bed", which in our case was a toddler bed that used his same mattress and familiar bedding, we dropped the outside rail and put a chair next to his crib. He very happily climbed in and out of his crib without incident. (We didn't want to take the rail off completely because we worried he'd roll out of the crib while he was sleeping.)

It's so hard when their little bodies grow faster than their reasoning capacities. I wish you the very best of luck with all of this (and because I don't think I said it already, congratulations on your new daughter!)