When used as an adjective, last-minute suggests occurring at the last minute, an expression for occurring very near to a deadline; very late in a process.
"At the last minute" is another form this takes as an expression, but remember, last-minute is an adjective.
Example: Last-minute additions to a contract. These are additions placed very near the end of a negotiation process.
Related: Eleventh-hour: Assumes a deadline of midnight. Something taking place very close to a deadline . Example: Eleventh-hour agreement.