A lower value part will typically have a higher self resonant frequency but it also has less capacitance and not as much charge that it can deliver. Sometimes many smaller value capacitors are used rather than a larger one.
On my boards I use a 68uF tantalum capacitor along with a 10uf ceramics, plus a couple of 0.1uF ceramics. The parts I use are surface mount parts that are mounted right at the ESP module. But I also have a 500mA low drop regulator mounted at the modules. Some people say they have had good results with 470uF. I find them to be bigger than I would like. The parts I use are relatively small.
There is a regulator, a 68uF and 10uF cap in the lower left corner of the ESP07 module. Right by the power input tabs of the module.
A couple of years ago I made some small PCBs that included some parts along with the ESP12 modules. Cleaner than the above but basically the same thing.
I picked 68uF because that is what I had at work. Same for that 10uF. The actual value is only part of the solution and not really a critical one. The right type of parts and the implementation is more important.
This is a board I designed for work. Same idea. Same cap right at the power input of the module.
I am aware of the delay issue required for wifi processing. In my sketches I don't think there are too many lengthy code sections, but I do have strands of up to 300 WS2812B leds that I control with the NeopixelBus and FastLed libraries that seem to work the best.
EVERY_N_MILLISECONDS( 10000 )
//connect wifi if not connected
if (WiFi.status() != WL_CONNECTED)
while (WiFi.status() != WL_CONNECTED)