DAFlippers wrote:A problem is the use of a linear regulator which will just waste energy and dissipate heat. Get some buck boost converters for your toolbox http://www.ebay.co.uk/itm/LM2596S-ADJ-3 ... SwKtVW0-6u.
If you do the math difference between what you suggest compared to a linear regulator you will find that they are the same as far as efficiency. Because of the low starting voltage you don't get much efficiency gain with a switcher and the switching power supply you suggested is horrible. Because of the voltage drop in the catch diode and the low output voltage the efficiency will be under 70%. A linear regulator's efficiency is 3.3/5=0.66. And if the input voltage was a little low, due to series resistances, then the switcher would have even worse efficiency and the linear would have higher.
The output of the linear regulator is constant where the switcher's output is a sawtooth waveform and the average voltage output is what you read with a meter, but the ESP8266 is sensitive to the lowest voltage available and that is the value that matters.
9.1.6 Output Voltage Ripple and Transients
The output voltage of a switching power supply operating in the continuous mode will contain a sawtooth ripple voltage at the switcher frequency, and may also contain short voltage spikes at the peaks of the sawtooth waveform.
As I said, that is a horrible switcher for low input voltage and low output voltage use. A mosfet synchronous switcher would be far better but it still is a switcher and still has upper and lower trip points so the above quote still will apply. There would need to be additional LC filtering to get rid of it.
A good low drop linear regulator is your best and easiest option. The regulator you used is claimed to be low drop but it really is marginal for a 5 volt supply input. While the voltage is listed at 5 volt the actual voltage can vary. And as stated in the first reply, series resistance due to the interconnecting cables may also be a factor.
With what you have shown you should be able to get it working properly with what I suggested.