@vaporlock
vaporlock, welcome to the forum.
Can you provide more information about the solenoid? A link to the datasheet or to the product description?
Also more about the power source. Is it batteries? What is it?
I'm an electronics novice, so the following suggestion is directed...
@everyone else
What about a large capacitor in parallel with the power source. (Large= a few hundred microfarads or maybe 1000uF. Like a photoflash cap.) My thinking is that it would sit at 12V and when the solenoid turns on it would provide a near zero source resistance for a brief moment to get things moving. If the power supply voltage drops a bit, the cap won't. Then once the solenoid engages, the cap would just come back to 12V until next time.
Do you guys think that something like that might "up the current?"
--tim