i think you should use a variable resistance at the output of the regulator in parallel with its output
so if the res. is large you can have the volage of the regulator
if the res. is 0 ohm the output will be zero
and the current of the regulator is not large to defect any thing
If you connect 2 silicone diodes (1N400X, 1N540X ..) in series with the output of the 317 voltage regulator the voltage will be reduced by 0.65V+0.65V≈ 1.3V and this "trick" ,in fact, shifts the output level from 1.3-16.3 to 0-15V ..
Theoretically this can work but the voltage drop over the diodes also vary with current. So voltage regulation may suffer between no load and full load. If you are not worried about a few 100mV change then it should be OK.
Another way of doing it if you have a negative voltage available is to reference the gnd end of the resistor to a min of -1.25V or more instead of gnd