OK, here is my plan:
A 7V @ 20mA booster from 5V, powered by USB.
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While connected, it will be charging a 6V battery.
When disconnected, the battery will be dropping from 6V to 4V
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The output will be boosted and regulated to 30V @ 1mA. This should be enough to run the quad op-amp LM324.
Notice that the single stage Larkin booster can deliver higher current, while the dual stages Hyde booster can deliver higher voltage. The Hyde booster has a nice smooth transient curve and less noisy as well.
Be patient with the links, it takes a few minutes to load and run spice.