Didn't check the TLS v.1.2 specs of these (edit: Firefox ESR 45 supports TLS v.1.2), but on 10.8. Mountain Lion I run Mozilla Firefox Extended Support Release (ESR) 45.x that will still work and still gets security fixes on 10.6, too. On the release plan I can see at least support until June 2017, though this timeframe for the last release could be extended or not. Some time ago it was announced last release for January 2017 and it was luckily extended!
Info on
https://www.mozilla.org/en-US/firefox/organizations/faq/
Download the ESR 45 release on
https://www.mozilla.org/en-US/firefox/organizations/all/
Then I've found, but didn't test:
NetSurf
Roccat
IMHO, Safari is such an old release, that it can't be really considered as a secure browsing option, even on 10.8. Personally, I think that 10.6 is a great OS and I wouldn't update to 10.7 as I can't see a real advantage in doing that.
Except you absolutely want to run the iCab Browser on it.
Another long term option is to find some actively maintained open source browser that will compile on 10.6 and compile it by yourself. Some starting point could be at
https://en.wikipedia.org/wiki/List_of_web_browsers
As there are projects like Classilla for Classic Mac OS 9 and TenFourFox for Mac OS X on PPC that are both based on Firefox ESR 45 code, there is the hope that ESR 45 will be maintained for a longer period of time or that there will be other projects arising that will somehow maintain browsers for Mac OS X 10.6 and lower.