So we have an RL DC circuit. If the inductor L1 has a high core permeability, there will be a high inductance and thus a long L/R time constant. So the circuit takes a long time to charge. Would putting a second inductor L2 in parallel with L1 help reduce this time constant at all, if L2 has a much lower core permeability (but is otherwise the same as L1)? If not, is there another way to reduce it other than decreasing L1 or increasing R?
Putting another inductor (of any inductance) in parallel with the existing one will reduce the total inductance, so the L/R time constant will be reduced if R stays the same.
If the objective is to get a specified current through a specified inductance and resistance within a short time, increasing the voltage will reduce the time for the current to rise from (say) zero to a given current. It does not change the time constant (which is the time for the current to rise to about 63% of the final current), but it makes the current rise faster.