Hi;
Since your input voltage is relatively small, may be you can use more powerfull LDO (like LM117 DPAK package)
But you should consider about the cooling necessities. As your logic core desing gets comlex, it draws more current and regulator may get hot.
Consider 1Amp@1.2V means a power dissipation on regulator aroud 3.8 Watts.
Or better way, use a simple stepdown like LM2575, and forget about cooling and that kind of problems (of course needs a little bit more PCB space bacuse of coils and other components).
Good luck