Inexpensive circuit architecture for power supply voltage adjustment?
It's really hard to do this with a generated voltage without degrading the voltage accuracy of the regulator.

But if you have a handful of digital outputs that you can dedicate to this, you can use a resistor digital-resistance-converter (DRC?) like this.  I've represented the regulator as an op-amp.







simulate this circuit &ndash; Schematic created using CircuitLab

The four trim resistors are switched out of the circuit by setting microcontroller pins to inputs and they are switched into the circuit by setting microcontroller pins as digital output lows. 

When all four pins are inputs, the voltage divider is R1 and R2.  When some pins are set, the divider is R1 and the parallel combination of R2 and the trim resistors.

I would suggest setting the lowest voltage with R1 and R2, the select the trim resistors to pull the voltage over the range that you want.  Do not choose all four resistors the same value.  Instead you will want something like a 1k,2k,4k,8k sequence so you can get 16 different values out.

For extra credit:


If your microcontroller has switchable high-side outputs (open-drain)
you can do that instead of switching input to output, then 
If your microcontroller has outputs as bits in a PORT register, arrange the
resistors with the largest value on the least-significant output bit.
Then with careful resistor value choice, you can write 0-15 to the
port register to get monotonically increasing voltage.