Good method to provide small additional voltage while on battery?
The pull-in and drop-out voltages of a solenoid are often very different. This is because the geometry of the magnetic circuit changes between open and closed. If you can persuade your 9-12v solenoid to pull-in, then you may find it doesn't drop out until the voltage drops to 5, maybe 4 volts, or even less. This is an experiment that it's worth you doing.

If you find the battery pack you are using is capable of holding the solenoid once energised (and only if), then you could employ a very large capacitor to give a switch-on boost, like this ...







simulate this circuit &ndash; Schematic created using CircuitLab

While the solenoid is off, C1 charges to the rail voltage through R1 and R2. When you want to turn the solenoid on, turn on both Q1.

Q1 pulls the bottom end of C1 up to rail voltage, which sends its top end to 2x rail, which applies this through D1 to the output. The capacitor is sized to store enough charge to pull the solenoid in before it discharges fully. Once it has discharged, the solenoid is held in by D4.