Current floor is 3/4 inch plywood subfloor with 1/4" plywood underlayment over that and then linoleum sheet flooring.
So the hardwood floor is up but as we all know the linoleum will be a bear to get up.
C'mon, man. As a DIY'er, thinking outside the box isn't just fair game, it's encouraged.
If push comes to shove, you can just pry off your baseboards and then pry up the old 1/4 inch underlayment with linoleum still attached. Then just put down NEW 1/4 inch plywood underlayment. And, if you want, install a nicer baseboard, too.
Alternatively, leave the baseboards in place and set up your circular saw to cut a 1/4 inch deep kerf. Go all the way around your room and pry up the 1/4 inch under layment inside the kerf (with flooring still glued to it). You can find any nails holding the last bit of underlayment down with a magnetic stud finder and dig out around the heads with a wood chisel and pull them.
As another option, rent a "toe kick saw" from any tool rental shop and cut the underlayment off 1/8 inch from the baseboard. (The underlayment still under the baseboards will just pull out from under the baseboards if you hook onto an end with a stiff wire.)
It just means when you put the new underlayment down, then you won't run it under the baseboards, or if you do, you'll have to slide it under them. And if the baseboard is 1/2 inch thick, you'll mark the edge of the underlayment at 7/8 inch from the edge to allow a 1/8 inch gap around the room for expansion of the underlayment panels (which the manufacturer prolly recommends) under the combined 1 inch thickness of the baseboard and drywall on the studs.
For about $15 per 4X8 foot panel, you're problem is solved. Ya just gotta start learning to look at problem from different angles.
And, don't use ordinary plywood for the replacement underlayment. Plywood is allowed to have voids in it's middle plies, whereas the requirment for underlayment is much stricter. The voids have to be much smaller or they have to be filled with water putty to avoid "soft spots" on underlayment, that can lead to problems with certain kinds of flooring (like ceramic tiles).
And, if this is a paper backed linoleum rather than a sheet vinyl, it's not all that hard to get the linoleum off. If you wanna go that route, post again and I'll explain the process. But, if you're not used to doing this kind of work, or fixing gouges in the underlayment caused by the flooring razor, then I'd recommend you pull up the old underlayment and put down new stuff. That'll give you a smooth clean surface to work with with the minimum amount of skill involved. If you're a relative newbie, that'd be the smartest route to go.
PS:
Here's how they make a floor:
In the beginning, there were the floor joists. They install the subfloor over top of the floor joists. Then they build the 2X4 walls on top of the subfloor. Then they install underlayment in each "room" so that there's a small gap between the edge of the underlayment and the bottom plates of the walls to allow for expansion of the underlayment panels. Then they do everything else, include hanging the drywall and painting. Then they install the flooring (carpet, say) over top of the underlayment in each room. Or, at least, that's how the floors in my building were built.
