A 4"w x3"h neck beam seems massive! Judging from that, I'm guessing that you're not planning to do a scarf joint for the headstock.
If you were to do a scarf joint, and didn't mind adding some wings to the headstock, for a 4-string neck-through you could go as small as 2.5"w x1.75"h (after squaring the beam up, and depending on body thickness).
On the other hand, if you plan things out carefully, something like a 4"x"3 (or a 2.5"x3") could get you two necks, back to back! Especially if the second one is a bolt-on and/or a scarf joint headstock.