The solenoid you linked doesn't list the inductance, but I am guessing based on my experience that it is somewhere between 100mH and 1H. It shows that it requires 5.4W @24V under DC conditions, so I=P/E = 5.4W/24V = 0.255A, which means that its DC resistance is E/I = 24V/0.255A = 94Ω, which is consistent with your measurement.
Since this is a solenoid coil, I cannot see any reason to drive it with a bipolar current. The magnetic force it generates either attracts a metallic object or it doesn't; the polarity of the applied current makes no difference...
Here is the reality of switching a 1H 94Ω coil:
I model the solenoid with the values above and a "worst-case" inductance of ~1H. I use a 24Vdc supply. I use a simple low-side voltage-controlled switch (to represent an NPN or NFET). I add a 1nF (guess) capacitance C1 to represent stray winding, wiring and collector-to-emitter or drain-to-source capacitance...
Note what the simulation shows: it takes about 50ms for the current in the solenoid to go from zero to near its final value of 255mA. More interesting is what happens when the switch turns off at 60ms. Note that because of the energy stored in the inductance while turned on, the switch is subjected to a damped oscillating voltage of +- 8000V as the energy from the inductor is dissipated in its own internal 94Ω resistance. Where can you find a NPN or NFET that will tolerate being subject to +- 8kV?
Fortunately, there is way to solve this problem: I add a Silicon diode D1 (called a "snubber" or "catch" diode) to the running simulation. Note what happens to the voltage at node C V(c), and the current through the solenoid, I(L1). Note that if "switch" is replaced with either an NPN or NFET, the voltage at the collector or drain is now well behaved. The rated breakdown voltage would only have to be a bit higher than the supply voltage..., and it never goes negative...
Note the shape of the current through the solenoid. It takes about the same time for the current to build up (50ms) as it does for it to dissipate after the switch turns off (also 50ms). Note that if you expect to "pulse" the solenoid on/off, the shortest period would be about 100ms, meaning that you can only drive it at at 10Hz if you expect a full on, full off cycle. Driving the snubbed solenoid with a higher frequency might enable using Pulse-Width-Modulation to adjust the average current through the solenoid.
In light of what I show here, what are you trying to do? Why do you think you need to drive the Solenoid current in both directions when a unipolar driver with a snubber diode seems to do everything?