What would increase magnetic field around a coil?

The strength of a magnetic field produced by a current carrying wire increases proportionately with the current through the wire, and it dies off as the distance from the wire decreases.

To increase the magnetic field around a coil you would either increase the current running through it (by increasing the voltage and/or using a less resistive wire), increase the number of windings, or wind the coil more tightly (bring the windings closer together).

The latter solution doesn't increase the overall magnetic flux produced, but it does make the coil more compact so the flux "leaks" less.