A friend has a Marlin 336A in .35 Rem, I think 1950 so same rifling as yours. We measured it at .361", so big bullets definitely the go. And don't rule out paper patching to get there.
Loading oversize is no sin. At the very least "fill the throat". I had a Mannlicher Schonauer 9x56 carbine with .353" groove diameter and .359" start of throat (and a long throat). It shot poorly with .354" bullets (#350447HP and 358009HP sized .354"). When I ran the #358009HP unsized at .360" it was suddenly producing 1.25" groups at 50 yds. Load was 40 gn 3031, velocity a little over 1900 fps, and the load shot to the sights.
Some older cartridges do not have a throat - the bullet just collides with a fairly abrupt rifling origin. This definitely applies to .30-30, with which I am familiar. I'm not sure about the .35 Rem, but I suspect it might also be throatless.
In the absence of a throat, I get good results from .30-30 by filling the neck with a .312" bullet. The neck takes on the guidance function that would be performed by the throat in a more modern cartridge. Put another way, in these throatless cartridges, use the biggest bullet that will chamber freely. That may require keeping a lookout for over thick necks, ensuring that everything has clearance.
One way of determining the biggest bullet you can use is to measure the O.D. of necks on your fired cartridges, then measure again when they are loaded with bullets of a known diameter. The difference is how much you can add to that "known diameter". Springback after firing will ensure neck clearance in this calculation.
Another method is to see how big a bullet you can get into a fired case without expanding it any further. Again watch for the odd thick neck that could give you tight chambering - or do a really gentle outside neck turn to touch just the high spots and to find the rogue cases.
You are only as good as your library.