In place upgrades between 4, 5, and 6 are not supported by RedHat and you risk impacting the support that you pay for by doing so. The only supported upgrade path is from 6 to 7. The expectation is that an upgrade path will continue in future releases.
You state that you cannot re-install but don't state why. Is it because the criticality of the system or is it because of the configuration of the application? I'm guessing the latter and that the system is not all that critical (because otherwise you wouldn't want to impact your support agreement). With that said, there is a way to upgrade and the procedure is provided by RedHat but, again, it is not supported by them.
The link is: In place upgrade