Linear regulator doesn't keep the right voltage when connected to ESP8266
The datasheet states a the minimum Vin must meet 2 conditions, one of them being:  

\$V_{in} &gt;(V_r + 3\%) + V_{DROPOUT} \$

which for a 3.3V regulator becomes
\$V_{in} &gt;(3.3V + 3\%) + V_{DROPOUT} =   3.4V + V_{DROPOUT}\$

So, that leaves 100 mV to be "used" for dropout.

You cannot use FIGURE 2-12 and FIGURE 2-13 from the datasheet to determine 
the dropout voltage because for these graphs the following applies:


  Note: Unless otherwise indicated: VR = 1.8V, COUT = 1 μF Ceramic (X7R), CIN = 1 μF Ceramic (X7R), IL = 100 μA,
     TA = +25°C, VIN = VR + 1V. 


And you do not apply Vin = 3.3V+1.0V to the regulator.

Moreover, the values shown in the graphs are typical values. You may happen to have an IC that deviates towards the maximum worst case dropout voltage. (For \$I_L\$ = 200mA, the worst/maximum value differs a factor 2.3 (!!) from the typical value.

I cannot find (yet) what the dropout voltage applies to this situation, but think an input voltage of 3.5V doesn't satisfy the condition mentioned as first in this answer.