Need to move backup folder
For backup folder :

If you mean the default backup folder as configurable in server properties --> database settings 



You can easily change that location to the one you want. If your SSD has only the backup files, then it should be ok to hot swap when your backups are not running. Just change the default location to the new one in the above stated location. A restart of sql server service may be required for the new values to get reflected.

A more easy and managable approach will be to explicitely specify the backup location in your backup scripts (or just use Ola's backup solution).

For LDF (or MDF) files :

You need to use alter database .. modify file to first modify the location in the system tables.

Then offline the database/s and then physically copy the LDF (or MDF) to the new location.

Once above is done, bring the database online.

See this answer for more detail steps by Mike Fal.