The PPA repository you added currently has 3.6.0 as highest version for Ubuntu 12.04 (on which Linux Mint 13 is based). As shared, you won't get a newer version of LibreOffice from Linux Mint (which uses the version from the Ubuntu 12.04 repositories), unless there is a security issue to fix. And then you would get a fix for the LibreOffice version as available in the Ubuntu 12.04 repository, not the one you installed from the PPA repository. Possibly the developers maintaining that PPA repository may update to a higher LibreOffice version, but that is outside our view.
Again, installing LibreOffice manually by downloading it from their website is the only real guaranteed way to get 3.6.3 now. But you will have to upgrade it manually for any new versions that become available.
With either the PPA, or installing LibreOffice manually from their website, you should be aware that security issues are not managed for the versions you have installed. Only the versions installed from the Ubuntu 12.04 repository are managed by the security team. So you should keep yourself appraised of any security issues with your version of LibreOffice, or at least be aware of the risk you take with this.
The point of a long-term support release is not
to be a rolling release where you continue to get new versions with new features. The point of a long-term support release is that once installed, you can be reasonably sure it will continue functioning on your hardware and that security issues are managed by the security team. For those reasons, you can continue to use your installation worry free for the full 5 years of the support period. If you want the latest version of software, which includes taking a risk it may not work without problems on your installation and will not be managed by the security team, then a long-term support release is probably not for you. I don't know what is so "must have" in LibreOffice 3.6.3 though, 3.5.4 works perfectly fine here