Can't help here as in I haven't tried it, but you wouldn't need to throw the other half away, if it is exactly a half dose you are wanting! Just keep it carefully in its little outer foil envelope and use it the next time. Personally, though, I wouldn't go down to a half once you've had a few weeks on a whole patch though. if it is the progestogen causing the side effects cutting it will also reduce the oestrogen - and would be better to try another product rather than cut it.
I would presevere for a few more weeks if you can and see how you feel then.
Hurdity x