There are few +1 games for Mutant Crawl Classics since the post-apoc OGL genre is small. However, finding a game to accompany MMC is possible.
My number-one pick is Goblinoid Games's excellent Mutant Future (MF). This one has been around a while, but it does not lessen the fantastic content you get here. Nearly everything is directly compatible, from low-tech gear to high-tech gear, monsters, rules for boats, robots, androids, and many more things packed into this book.
You need to do a descending AC conversion for all the monsters and gear, but it is 19 - MFAC = MCC AC.
The mutations in this game are perfect for monsters or other creatures, where you just want an "attack ability" and nothing for which you want to roll on a mutation chart. For example, an energy ray mutation here does a 4d6 beam of energy (roll or pick type) to 50' every three rounds. This is perfect for many monsters and works well with MCC as a creature power.
MF is a worthy game to play on its own, and if you want more for this game, pick up a copy of the old OSR standby Labyrinth Lord (LL). Use most everything out of here (except magic), rename the monsters, and give them a few random mutations, and you are all set. MF plus LL is a one-two punch of extra stuff for MCC and considerably expands your game while staying in the OSR.
The MF+LL combo is also a good "fantasy baseline" for MCC games, as Mutant Future assumes a Middle Ages level of technology in future societies. War is still made with siege engines and sword fighters, cities have walls, and the general level of technology is deficient, with most ancient artifacts forgotten or useless. Travel is done with carts pulled by mutant animals or boats on a river. There are no "Mad Max" cars here, and the knowledge of the ancients is more removed from everyday life.
There are rules for gunpowder firearms and cars in MF (the cars lack AC values, but that is easy enough to guess using the armor types), and the guns are an excellent addition, so you could hack in a "Mad Max" style game here.
There is also another option for a +1 game, the PoD versions of the original Gamma World, but only the 1st, 2nd, and 4th editions. I recommend 1st or 2nd. Note that the second edition has many more fantasy inspirations, and the layout and organization are terrible, but it has a complete set of monster art and lots of "stuff" to use. The fourth edition is like a cleaned-up second edition (with much more stuff), but they introduce many more rules incompatible with B/X, and the writing in the book is very long-winded.
Overall, Mutant Future is easier to use as a quick +1 game for MCC, but Gamma World is iconic in many ways. There is a massive tone shift in Gamma World, where GW 1st is more like Mutant Future, and GW 2nd is more like MF+LL with many converted-in-fantasy elements. The fourth edition of GW feels more like the early Fallout games.
As the editions were published, Gamma World generally got worse as it got farther away from its B/X roots, and they tried to make it a superhero or sci-fi game. The first edition has the best tone and setting, the second introduces iconic monsters, and from there on, they just keep adding tech and superpowers. Later editions of the game also got unwieldy in the lists of stuff they provided, and the organization could have been better.
The fourth edition was based on AD&D 2nd, but the game developed new rules and subsystems it did not need. This edition also needs to be revised. Instead of keeping something like mental attacks simple, they spend two paragraphs laying out rules about plants being only able to mentally attack plants and how robots and AI are not affected by mental effects. Instead of keeping it simple (plants, humans, mutants, cyborgs, and animals have brains and can be affected; computers and robots cannot), they spend two paragraphs laying out rules about how plants can only mentally attack plants, how mental attacks on machines fail, and so many other nuances I wish an editor with a laser-chainsaw would have spent more time here.
Many of them followed the desktop publishing fads of the day, overusing two-letter abbreviations to the point of nonsensical notation. They tried to look "cute" over having a readable book.
Mutant Future is a "reset" on the genre and gives you the best Middle Ages setting to use as a baseline society. I like basing tech and knowledge on a fantasy baseline since this lines up with 1970s fantasy inspirations the best. Once you enter the 1980s and 1990s, the games start copying Mad Max, Fallout, and other "pop culture sensation of the week" sources.
One problem with the 2nd edition Gamma World was the monsters; they have fae-like "Lil" and a massive list of powers in the setting. In our games, these became the only fae, when in the original spirit of the game, taking a fae as a baseline and applying a set of random mutations to it to create a unique species of fae is a much better option and keeps the game fresh.
Perhaps there are "glow fae" that have radiation powers and live around ancient reactors since they are immune to the energy and feed off it. This system is more like the MF+LL combo, where you pick a baseline fantasy monster, roll random mutations, and see what you can imagine.
The GW 2nd monsters can become far too iconic and take over the game like licensed IP, and it gets predictable and boring. I love the 2nd edition monsters, but in my version of the world, every monster or local species should be unique and different. You don't know if that mutated carp will shoot radiation beams, freeze the water, or try to take over your mind.
The world is much more unpredictable when you see something and have no idea what it can do, rather than repeatedly seeing the same old "rad fish."
All that said, using Gamma World as a +1 game for MCC is only for advanced referees with knowledge of the editions, the problems they introduce, and a love of the source material. The Mutant Future game is much more straightforward and easy. If you want the fantasy elements, MF+LL is far easier than GW 2nd and gives you infinitely more flexibility.
Also, the LL+MF combo is worth as playing as its own game, too. Even without DCC, this is a highly compelling combo that gives you the best of both worlds.
Also of note, Kevin Crawford is developing Ashes Without Number, an update of his old Other Dust game, and a game in the same genre as Gamma World, MCC, and Mutant Future. Any of the ...Without Number games are excellent, top-quality toolkits, and it is worth keeping an eye out for this one.