How large can the bitcoin salt value be? To mine bitcoin I believe that you take the input and add a salt number to that then run the hashing algorithm to find a suitable output, but how large can the salt be, couldn't I just take the current input subtract it from the previously found bitcoin input+salt to get a new salt, and use that new input plus new salt to find a suitable output (same output as the old bitcoin)?
It's actually called a "nonce".

It is a 32-bit number so is maxed at 2**32 or 4294967296.

If the hash and data given are valid, you can always hash the block header data to get the same block hash.  If anything is changed, including the nonce, the hash will change, invalidating the block (header).

Each valid nonce is dependent upon the data being hashed and the target difficulty according to the protocol.