Only amateurs would choose to implement a "Triple AES", so it is very likely that they will also write a buggy implementation. Triple DES has not been used because it was a good strengthening method, but only because it could be used with unmodified hardware modules designed for simple DES. When a cipher strengthening is done in software, there are much better methods.

The best way to strengthen AES above the standard AES-256 is to double the block length from 128 bits to 256 bits. Increasing the key length over 256 bits is much less useful, because the key length is not the weakest point of AES-256. A 256-bit key is strong enough even against quantum computers, but short 128-bit blocks can be a vulnerability in certain applications. The key schedule algorithm of AES, which converts the cipher key into a set of round keys, is mediocre, so the length of the cipher key is the least important concern about the strength of AES.

The original Rijndael proposal had a stronger variant with 256-bit blocks, which has not been retained in the standard. Nevertheless, it is easy to implement it with the Intel/AMD AES instructions or with the Arm Aarch64 AES instructions. Intel has even published an application note describing how to do this, when the AES instructions have been introduced in the Westmere CPUs.

After increasing the block length, increasing the number of rounds can provide additional strengthening. Another choice would be to replace the standard key schedule algorithm with a stronger non-standard algorithm (i.e. one providing more random round keys). Increasing the key over 256 bits provides a much less useful strengthening in comparison with the cost required for executing the additional necessary operations.

SPHINCS+ / FIPS 205 should be available soon.

FALCON ...unknown FIPS draft TBA soon.

These are newer quantum resistant algorithms, and should be considered in your future maintenance cycle as they become available in the libraries.

NIST has some of the brightest minds in the world. When they suggest something, than one should probably take the advice very seriously. =3