It usually should but sometimes it does not.
Would suggest getting another External Drive without least 2 1/2 Greater Drive Capacity of the Internal Drive and start a new TM Backup.
Keep the Older Drive is a Safe Place ( Safety Deposit Box perhaps ) incase you need to use it again to restore some files from it.
That or the real issue is the Internal Drive does not have enough Empty Space to store all the Snap Soots that TM Backup has been waiting to transfer to the existing Drive .
To see all the exiting TM Backup Snap Shots on the internal drive
Terminal command
\tmutil listlocalsnapshots /
The Terminal will show a list of local snapshots with names