All of this should be read with a very large "I AM NOT A LAWYER" hat on. The very fact that you've asked this question shows that you have sufficient doubt - you should either consult a legal professional or just take the easier (and cheaper) route and live with the GPL.
You, as the author of code, own copyright to the code you author and can license it as you wish - this is the entire legal basis that allows the GPL to exist in the first place.
How does that apply to your example? It depends on the modifications you make. If you've made some modifications here and there to existing code then the current license should stand; this should apply no matter how extensive the modifications are.
If you have added entirely new code of your own, if that code lives in it's own modules, and if it can be reasonably shown to be standalone (or at worst capable of standalone functionality), then I believe (IANAL) that you actually do have choice of license for that code, and that code alone.
Existing work to support this would include incorporation of e.g. zlib code into GPL projects. That's the level of "standalone" I'm talking about here; if you're utterly satisfied that you're watertight on this (and if you find yourself having to try prove it then you're probably not) then this should apply.
Even then, the GPL does require that the license you choose be GPL-compatible, so you can't just pick and choose and you definitely can't keep your own source closed. The FSF site maintains a list of GPL-compatible licenses, so it's wise to restrict any choice to licenses from that list.
The wording of the question "Am I required to claim a copyright on my modifications to a GPL-covered program?" in the GPL FAQ supports this reading of the situation: this is an explicit acceptance from source that changes you make can be placed in the public domain. See also "Can the US Government release improvements to a GPL-covered program?", and - quite explicitly - "If I add a module to a GPL-covered program, do I have to use the GPL as the license for my module?".
Note that in all of these cases the entire program itself must be under the GPL, there is just allowance for individual parts of it to be under different licenses. Note also that these are just FAQ answers explaining the intention of the GPL; if this were to be tested in court it would be the actual license text that was tested.
Finally, and it should be quite obvious, under no circumstances can you re-license existing GPL code; once again: it would have to be entirely new and entirely standalone.
That's a hell of a lot of "if"s for an answer, so the path of least resistance is most likely to just swallow your dislike of the GPL and use it.