You may have to settle for as recent of version of Firefox, as your OS X 10.6.8 will allow.
Even an older archived Firefox would be more recent than Safari from that era.
https://www.mozilla.org/en-US/firefox/new/
see:
https://support.mozilla.org/en-US/kb/how-download-and-install-firefox-mac
The method to get a later version of Safari, would be to see if the computer can be upgraded
to a later fully new OS X which includes a period-correct Safari version. That's about it.
And an upgrade to a later OSX may compromise the kind of computing experience you'd
found acceptable with Snow Leopard 10.6.8. I'd consider a later computer, as a vehicle to
access later OS X versions, and keep the older computer (with backups) in Snow Leopard.
The online App Store may suggest a later OS X version than you have. The latest it may
(or not) support is Yosemite. That would likely require a RAM upgrade and perhaps a new
hard disk drive of faster rotational speed, or perhaps an upgrade to SSD, to add speed to
allow the bulkier newer OS X to work OK. Then you may need to get newer applications.
Other available OS X versions, not as new as Yosemite, would be available by purchase
in the Apple Store online. A code would be sent to your email, to unlock these downloads.
If you fall into the potential upgrade trap, be sure to create viable external clone or bootable
copy of your current system on external device; and have backup Time Machine archive, too.
In any event... there isn't a newer Safari for Snow Leopard 10.6.8.
Good luck & happy computing! 🙂