how to increase the speed of a solenoid?
You can drive the solenoid from a higher voltage than its rated 12 V to bring it in with greater force and therefore more abruptly. The solenoid will produce an increasingly strong magnetic field as the core is pulled in, as it gives the inductor an iron/steel former instead of an air former. So, once it's pulled in, you can reduce the solenoid drive voltage back to your 12 V to hold it in place.

One way to do this is using a switched supply/regulator in your drive circuit. However, a concern you will have to resolve is ensuring your drive circuit does not have a failure mode that continually drives the solenoid from this higher voltage. Depending on your application, this can burn the solenoid out or even cause a fire, so it represents an electrical safety hazard.

I had an application using several solenoids that required sharp pull-in and on for only a short duty cycle of a machine's operation (0.5 sec or so every 1.5 secs). The solution I used was to charge a large capacitor when the solenoid was off then dump it into the solenoid when it was energised. The profile of the capacitor discharging into the solenoid nicely covered the time to pull the solenoid in and gave it a nice sharp action. It was very reliable and we made a large number of them. We had tight cost constraints but, more importantly, the circuit would have to be safe under Single Point Of Failure (SPOF) conditions (any component short/open circuit) to meet safety approvals. My circuit met that, as once the capacitor's discharged, there's nothing to overvolt the solenoid with. The current requirements could be handled by our existing PSU as the capacitors were charging while the solenoids were off, unlike a higher-voltage PSU approach.

(Excuse the rough'n'ready schematic)



The exact capacitor value depends upon the solenoid and time is needed to recharge it during the solenoid off-time. I used 470 uF but in guessland I would start with 100 uF then 220 uF with your solenoids. Trial and error in the real system under real mechanical loads was the quick and pragmatic way to choose our capacitor values, our loads had springs and all sorts. Charging resistor Rc can be calculated from the standard capacitor charging equation to ensure it can charge the capacitor in time from a practical resistor value. By practical, I mean an Rc the PSU can supply the initial C charging current for (something like 11/Rc amps) and of a suitable power rating. For ours, this meant a 5 W resistor. I used 1N4001 for all diodes.

Note that I used a power op-amp back then but you may well favour a FET-based H-bridge driver. I showed the op-amp because it also illustrates the control signalling, which will be different with a different driver.