Hi,
One of the main problems in the transfer of energy through free space using a changing magnetic field is the permeability of free space is very very low. Permeability in a material is like conductance in a conductor, so low permeability is like low conductance so it is hard to transfer energy through the medium. So free space is like a poor conductor of magnetism.
In a poor conductor everyone knows that to get more current flow you need a higher voltage or a wider cross section. In a 1k resistor with only 1v applied we only see 1ma flow, and thus only 1mw of power. But increasing that to 1000v we see 1 amp, and that's now 1000 watts of power. Quite a difference. If we incrase the cross section we may bring that resistance down to only 1 ohm, and thus 1v across 1 ohm gives us 1 watt and that's still 1000 times more than we had before.
A magnetic field is three dimensional, and the cross section is two dimensional, so there are really two dimensions to the magnetic problem also. One is the field cross section, which increases with an increase in coil diameter, and that is similar to the poor conductor. The other is the same as in the voltage/conductor case: a stronger magnetic potential. Either way the transfer of energy goes up.
The B field is related to the H field and the H field goes up not just with more current but also with more turns on the coil, so to get a stronger magnetic potential we need a higher current and/or more turns on the coil.
If the coil ends up being 'long' (front to back) then a magnetically active core would probably help because it would help to get the coil turns in the rear of the coil magnetically closer to the front of the coil and thus reduce the distance from the back of the coil to the target, making it similar to the distance from the front of the coil to the target.
So in the end we see three for four things that can increase the power transfer:
1. Larger coil cross section.
2. Higher current in the coil.
3. More turns on the coil.
4. For long coils (front to back) try a magnetically active core material with high permeability.
If the receiving coil includes capacitance for reasonance, then the transmitter would have to match the resonant frequency for best results too.