How to get a LDO voltage under 1.1V with lm1117 adj LDO?
You could use an LM7705 and three tiny ceramic caps to generate a -0.232 volt negative rail and then you'd be able to get down to about +1V. 

\$V_{OUT} = V_{REF} (1 + \frac{R_2}{R_1}) - 0.232 +I_{ADG}R_2\$

So for R2 = 0, Vout ~= 1.02V 









simulate this circuit &ndash; Schematic created using CircuitLab

Where \$R_2 (\Omega) \approx 96.2(V_{OUT} -1.018)\$



Or use another type of regulator (what I would suggest if the neg rail isn't otherwise useful).