I would put the 2013 laptop into Target Disk Mode. You would need to use an Apple USB-C to Thunderbolt 2 Adapter along with the necessary older style Thunderbolt cable to connect the adapter to the old laptop.
You can also enable File Sharing on one of the Macs and transfer the files across the network.
Set up file sharing on Mac - Apple Support
Or you can connect an external USB 3 SSD to transfer files between the two Macs.
Or you can take advantage of iCloud file syncing so that files are accessible to both systems.
Set up iCloud on your Mac - Apple Support
Just make sure to have frequent and regular backups of your computer and all external media (including the cloud) which contains important & unique data. Cloud file syncing is NOT a backup.