Louder Horn / Circuit / Different Speaker to be louder
Is there anything I can do in my circuit in order to make the horn
  louder other than using a higher voltage which is not an option?


No, there isn't. With a 24 volt peak-to-peak square wave into the voice coil, that's the same as feeding 12 volts DC into a 6 ohm resistor, which means that - under those conditions - the power your voice coil is dissipating is 24 watts. 

Also, there's a problem with the data sheet, since that 1 ampere current limit translates to 12 watts, not their specified 20 watts max.




  Will going to a 4 ohm speaker be louder than an 8 ohm speaker in this
  case?


Maybe, maybe not. The speaker's impedance doesn't determine how loud it is, what matters, basically, is the geometry of the horn, how far the diaphragm can flex, and how much current you can put into the voice coil before it melts.