I've done .45-70 using an 1885 Remington-Lee magazine. Works way better than the Navy Arms mags. The SMLE is a descendant of the Remington-Lee; the Lee magazine snaps right in and feeds properly with no fettling needed. You have to fill in the sides of the magwell to keep it from rattling side to side, though.
.444 Marlin will work with Saiga-410 magazines. Some mags will work perfectly, others will only hold 8 rounds, or have feed issues. You don't know until you try. .405 Winchester will also work through some Saiga mags.
Your less-traveled option is .375 Flanged Nitro Express 2-1/2", which is basically a blown-out .303 case with a .375 bullet. BSA used to offer that chambering in civilian-market SMLEs. The .375 will work in the 5-round sporter mags; I'm not sure if the 10-round military mags will accept a full 10 rounds. You can get .375 brass, and Kynoch still loads .375 NE, though the new-production stuff is apparently for double rifles and has the bullets set out too far to go into the SMLE magazine.
A small-bore option is the 6mm Musgrave, which is just another of the many 6mm-.303s, except it's CIP listed and you can buy commercially-loaded Musgrave ammunition. Lee Precision has dies for reloading. You're not going to get benchest accuracy out of an SMLE, but you can tap into the high-quality 6mm barrels and bullets for target shooting, and there's a fairly good selection of 6mm/.243 bullets for hunting.
There are a bunch of tiny feed ramp cuts in the SMLE receiver to assist the .303 cartridges in their journey to the chamber. In general, they don't affect fatter cartridges one way or another, though there might be differences among SMLEs. The small bolt diameter and low barrel-to-magazine-lip distance do a pretty good job of holding everything in alignment without needing a bunch of fiddling.
The Enfields use a flat bolt face, a long-travel extractor hook, and a little screw through the left side as an ejector. Generally they need no work whatsoever for any rimmed cartridge, and they'll work well with many rimless ones as wall - 7x57 Mauser was once a popualr conversion in Canada, and of course some rifles were re-arsenaled into 7.62x51 NATO.
7.62x39 conversions were popular a few years ago, using AK magazines. There have been some homemade .308s, but I think that's asking a lot of an elderly rear-locking action. .300 Savage will generally fit and feed through anything that will run .308, and give you most of the performance with a lot less pressure.