I actually don't access my email from my phone, but the hosting provider (again, pair.com) has a web UI for accessing email (very roughly comparable to gmail) so I certainly could log on there, or could install an IMAP email client on my phone to browse the email until it gets POPped off the host when I am back at the laptop.
When switching laptops, I would restore my local email from a backup (I guess I'm willing to lose a day or three, and try to remember what was important), or rsync the email files back from my fallback/old laptop. I have a somewhat regular routine to upload those files to my pair.com shell account storage (tar, gz, gpg encrypted, then scp, and sometimes to my free google storage and/or AWS glacier), as well as rsyncing key files with a script (which has grown over time to become a full restore for all the accounts I use on my computer) to the old/fallback laptop, and semi-periodic full backups (carefully selected then tar/gz/gpg encrypted) of files that matter, to optical off-site media (not enough volume for tape quite yet I guess).
I tried to answer the access question better in another edit to my original comment above.... Let me know if not.