Nice. I love PDF format you use, and the way you labeled V and I on your schematics. I would do the circuit differently (see below), now that you & JF figured out the jack switching and posted it in the schematics.
Ed
+3 +-----------------------+ | | --- +---[317]---+-->|---+ D1 isolates the 3V battery from R1-R2 - | | | D1 | to gnd, and the 317 from feedback from --- | | [R1] | the load, assuming a cap across the load - | | | [Load] to keep it powered during switching. | | +-----+ | Set Vout to 3.7, assuming a .7 drop in | | | | D1 at 15 mA. Use R1 = 240 and R2 = 470. | | [R2] | Or, replace R2 with a 500 ohm pot and | | 9V Gnd | | adjust until voltage to the load is | +---o o-----+-------+ 3 volts. | | | |>|