Obviously you can do $n=16$ pieces just by splitting the kites into drafters, assuming flipping is allowed which it generally is with 'hat' constructions. A drafter is the 30-60-90 triangle you get when you dissect a kite into two congruent pieces.
Furthermore, you can use four drafters to tile a 'doubled' drafter. So that gives $n=64$ as well, in fact n can be $16*4^m (m=1,2,3,...)$ by dividing drafters into smaller and smaller pieces.
And as Jaap points out, you can use a great many triangle divisions, not just 2^2. If we write down just the new ones, ie avoiding overlaps, we get $4, 8, 16(p*p)^n$ where p is primes starting at 2 and n is all positive integers. And with OP's division into four 30-30-120 triangles, add $2*16(p*p)^n$ and since those are symmetric triangles, add $4*16(p*p)^n$ as well. Some overlaps there but I think both terms add new numbers. Possibly there are further divisions of kites and kite-pairs into triangles or other simple shapes as well.
I see no dissections into 2,3,5,etc shapes (yet). But I can see plenty of searching to fill in some of the larger missing numbers.
Actually I realise I've missed out a bunch of possibilities. We should write $(1,2,4)16(p*p)^a(q*q)^b(r*r)^c,...$ where $p,q,r,...$ are primes starting at $2$ and $a,b,c,...$ are non-negative integers.