esp8266 power supply problem
This answer is to show my ESP8266 power-on plot. It is provided to compare against originator's trace, so it is possible it is not an ultimate solution.
My setup consist of 5V 2A power supply, connected to AMS1117-3 with some additional capacitors.
Capacitors: on input of AMS is an tantalum 10uF and output of AMS - 220uF electrolytic and  close to the ESP8266 Vcc and GND board terminals is located a 100nF capacitor.

Scope trace:

Scope is on 500mV/div, so overshoot is around 250mV.
I would suggest to use capacitors to suppress initial voltage overshoot. Looking at the originator's scope trace it would be beneficial to use small capacitor (100nF) as close as possible to the ESP module.
Also, as is mentioned, using non-original AMS1117 may bring additional problems.
EDIT:
Some wording has been changed to make my post clear.