heat transfer is basically influenced by the following things:
1. Area of heat transfer. the bigger the area, the more heat is transfered.
2. temperature difference. this is the driving force of the process. the bigger, the better. =)
3. heat transfer resistance. (implemented through the use of a "heat transfer coefficient"

this is largely dependant on thermo properties of involved substances.
getting back to your question with these 3 influences in mind, we can deduct the following:
does the area increase? this is the case if your spiral is a tube/pipe (whatever is the correct word

wound around the vessel. more windings from the same pipe means the spiral has less free (unused) space, area is getting bigger, heat transfer is improved. you will also get a different LMTD with a longer coil (which is essentially the effect of more windings - longer coil). as the pipe/tube will be the same nonetheless, the flow regime doesn't change, provided the heating/cooling fluid flow doesn't change. so you have better heat transfer.
if you have a double-walled-vessel, and the "spiral" is made by a spirally wound obstruction between the two walls, changing the spiral would not result in bigger active area. It would change the fluid properties, once again, provided the cooling/heating fluid flow is not changed. due to the change in cross area of the flow path with the new spiral, the flow regime, expressed through reynolds, changes, which influences heat transfer - again to be better than before. the more turbulent your flow is, the higher is - generally speaking - your heat transfer coefficient. this will once again have an effect on LMTD, as more heat will be transfered over the same area. so in this case, more coils will once again help you.
the things said above are from a purely heat transfer side of view. if dp in the cooling/heating jacket is of importance, this will certainly change with modification, and to the worse. (you will have more dp).
also, the increase in coils will have no real effect if your overall LMTD is too low. at least the effect will not be worth the effort.
in addition, try to raise the heating/cooling fluid throughput to give yourself better performance without neccessarily changing the apparatus itself. this seems to be a more economical solution.
hth,
chris