If the nightly DIFF backups are not too big:

Take a FULL backup earlier in the day

Restore the FULL backups on the new server WITH NORECOVERY

Keep running your TLOG backups as normal

Right before cutover time:

Run a final TLOG backup then disable the TLOG backup job.

At cutover time (for each user DB):

Alter the DB to SINGLE_USER mode, take a DIFF backup, alter the DB to OFFLINE

Restore the DIFF backups on the new server WITH RECOVERY

(note: you can stripe the diff backup to speed it up)

If you need to backout then alter the OFFLINE DB's back ONLINE.

You can test this ahead of time to have your jobs/scripts ready to roll and know how long it will take (just don't alter the live DBs OFFLINE).