First draft should always be to add local capacitance at the power pins. Simply take a 100uF capacitor and place it near the power pins of the offending module.

Also, be sure to add 0.1uF decoupling near any other potentially sensitive circuitry power leads.

When all else fails, perhaps try a hot-plug circuit strategy. This usually involves a P-channel MOSFET with resistors and caps arranged around it so that it turns on slowly, limiting inrush. You can google and find a half-dozen examples.

Here is one: https://www.maximintegrated.com/en/app-notes/index.mvp/id/2158

First draft should always be to add local capacitance at the power pins. Simply take a 100uF capacitor and place it near the power pins of the offending module.

Also, be sure to add 0.1uF decoupling near any other potentially sensitive circuitry power leads.

When all else fails, perhaps try a hot-plug circuit strategy. This usually involves a P-channel MOSFET with resistors and caps arranged around it so that it turns on slowly, limiting inrush. You can google and find a half-dozen examples.

Here is one: https://www.maximintegrated.com/en/app-notes/index.mvp/id/2158

If the problem is bulk-capacitance, going to a higher voltage will cause higher currents to be drawn on the 9V rail, causing even higher currents on the 5V rail. If you are feeding a switching converter, then bumping up to a 9V rail might solve your problem as the switching converter will fold back a bit.

Overall, I would try it in the order presented:

1. more local capacitance
2. soft start circuit
3. boost regulator

Good luck!