The bottle jack on springs type are indeed miserably wobbly. Even if you make great efforts to guide things. I saw one YouTube contribution that repurposed a set of motorcycle tele forks with the bottoms of the sliders cut off as guides. A for effort. Not sure if it was worth it. My home brew has a 2 1/2" diameter pusher in a 4" deep guide which I expected to work but, in practice I had to open up the hole a bit to provide clearance as it tended to jam when a nice running fit. Much better than the usual but not as good as it ought to be. Running the pusher in its own bearings with no mechanical connection to the jack is probably the only way of getting true stability.
If you only have small work in mind consider building something around one of the "12 ton" hydraulic rams used with various puller kits. Can be got inexpensively as spare parts from E-Bay, Amazon et al. I imagine something taking components in the 4 to 6 inch wide range should be practical. Thick bottom plate with suitable hole for to locate the support and for things to go through when needed. Thick top plate threaded for the ram. Two joining rods and Bobs your mother brother.
I faked a 3 and a bit inch capacity one up very quickly using the ram and slotted bar unit out of my big hydraulic puller set using studs from from my inexpensive import milling clamp set for tie rods. My base was about 3/4 inch thick. Found a slice of round steel bar with a hole in it out of the come in handy box. Drilled and tapped two short lengths of 1" Ø steel off cuts to screw the tie rods into. Milled some short flats on the sides to give a decent register and welded them together. Used standard nuts on the top of the tie rods with hefty T style washers to guide the rods properly through the slotted bar.
Did what I needed. Maybe I'll make a proper one someday but nowt so permanent as a temporary job. If you are short of space a device of this style is easily dismembered and packed away. Proper presses, even arbor ones, take up a lot of space and don't really justify the room given their relatively infrequent use. But when you need one you need it.
The (relatively) inexpensive universal push and puller bush replacement kits and bearing insertion pusher kits are are a good source of push and support accessories. Probably not worth buying for this purpose alone but if you can justify for car fixing duties go for it. After years of rooting around for bits that might do I'm so happy that I bought both for car jobs.
The monster G-clamp style pusher thingies sold for automotive ball joint removal look like they might also do for an improvised system. Unfortunately they deflect under decent loads and cannot be relied on to push straight.
Clive