(This might be increasingly-useful through 1.16's planned Nether Update)
So you’ve spawned in the nether through the portal and want to ride back home before your variety of horse gets lost. Or you’re trying to repeatedly get the not-quite pet back safely-enough that you might not need a diamond pick(axe), every time.
Since the player and other mobs tend to seem to spawn in the middle - and the more range, the more middle away-from portal edges there is, potentially - of the nether portal when-portaling. This might help.
I find that the minimum 3-high by 2-wide internal portal frame (5-high by 4-wide external - total - obsidian range) is too narrow for safe-enough mob-movement, through it, especially when they’re bigger! So varieties of horses - which are all at-least 1.5 blocks wide both-ways, 2 I think in one - I’ve even had apparently-die then despawn, in holding areas of anything-less than 4X4 internal block(s’ ) spaces.
Which’s made me build 4X4 (so they can be ridden - dismounted of course to portal - and the player then portals separately, too) - no head-hitting! (one less threat in the nether, “a gimmee”) - rooms sideways, portals vertically, and so always these flanking the nether portals. Empty space for more natural and original portaling-in, movement.
A further use of the extra room is player movement - including to push-around mobs, especially again types of horse-like things - the smaller they are, the more of them can get stuck inside, and the less individual mobs can be able to have as much surface to push, as many of them out. Otherwise, be ready always in any case possibly with a diamond pick with high efficiency (so iv /v best), to take at least 2 blocks out, which generally can be done once getting visibility by leaving the portal distortion.
There’s also an issue where the portal itself blocks the interacting with things further than it, so sideways-breaking blocks is an issue even when standing in the portal frame (and distortion of vision), when attempting to break obsidian say past additional sideways portal part. Again, the empty space implies sideways movement is clear, so another point is to keep it all - on the same - level and this can allow for best movement /pushing of mobs (and all of this least effort to break 2-4 blocks before the mob suffocates in the obsidian).
I used to lose about 1 /4 of my especially-horse variety mobs, going-through the nether, because of almost always - attempted - portal(l)ing. Ever since, it’s less than 1 /16, mainly due to other stuff (fire from the 4 sources, and drops of course always hurt more unpredictably).
This can also be done by riding mobs wholly-within the nether, though boats on (the non-melting) ice (so not regular, but packed or blue) are faster in the longer-term (the more “expensive,” the faster), which I could call tobogganing. Another tip of course on the nether side is to pre-vent the ghasts’ explosions from ever reaching the nether portal /storage chests /desired mobs including-player, there.
So wrapping-sections around (giving me 6X6 on the outside, with more height for where stair’ing is required, as I just build in the open areas, generally though of course anywhere’s fine and the more behind blocks one is, then hostile mobs won’t - heh - fire) to keep from a line-of-sight, to hostile mobs, accomplishes this. One wrap for each desired - separate - protected area, with the roof ladder too in a safe place regarding protection, while building with non-(in)flammable material is also pretty much a must.
If nothing else, like nether fortresses, it’s good to have available-supplies (flint-and steel) for re-lighting the nether portal - or expanding via lighting another, elsewhere - handy, and while flint-and steel is very useful. It’s anyone’s decision as to whether to use an ender chest slot for such storage.
Another possibility is leads. They may lead your horses out, but may not (just like - attempting - pushing through blocks-remaining).