People don't often do proper backups of their computers (especially their laptops).

I'd like to suggest that you do a full backup of the old laptop to the external drive, with file verification. Then, any file you later need will have been backed up for archival purposes. You can then restore some or all of the files to the new computer.

(I use the version of Retrospect that came with my external USB drive, but any backup program will have options to efficiently restore files without prompting.)

And, please be sure to back up the new computer regularly, too.