The proposed method would have the advantage of keeping existing known abbreviations, which are _already_ less secure than SHA-1, while keeping the security of the second hash.
It also has the disadvantage that the full hash would become excessively large and unwieldy, so pros and cons.
This solution would basically just make the UI backward-compatible while still requiring the complete modification of the internal to change the hash function.
You'd still risk a collision if you refer to commits using a shortened hash outside of git but something tells me that you don't even need a vulnerability to take advantage of that if you have an attack vector. For instance github seems to use 7 hex digit in short hashes, this could probably be bruteforced relatively easily (be it for SHA-1 or SHA-256). To give you an idea I looked at the current bitcoin difficulty (which AFAIK uses two rounds of SHA-256 internally and works by bruteforcing hashes with a certain number of leaning zeroes) and the hashes look like this: 000000000000000000028048b31e42bd53d3b36da90d1a840ae695ec1a5ee738
Note that git doesn't concern itself with reversing a hash function. The commit contents are part of a repository, there is no value in guessing the commit contents basing on its hash. Here, the hash function choice is purely about collision resistance.
But yeah, don't do weird things with hashes. Cryptography is hard. Don't invent memecrypto: https://twitter.com/sciresm/status/912082817412063233, it's not going to increase the security. Use a single algorithm if you can. Don't transform the output of a hash function in any way.
Notwithstanding, i still dont like it as an idea.