There is no "inrush", per se, as is found with AC coils. Instead, the current changes in several distinct, but observable, increments:
a) upon initial application of voltage, there is an exponential increase of current to the point when the armature starts to move (pick-up).
b) as the armature begins moving it generates a back-emf opposing the applied voltage.
c) as the armature accelerates the back-emf increases causing the current to drop somewhat below the level reached in point b), above.
d) When the armature is fully seated the coil becomes more inductive than when it was open.
e) This results in a slower growth of current until it reaches an upper limit.
f) Finally, as the coil heats up, its resistance increases, thereby lowering the current to a final value.
g) for most coils, at least those which were connected to our Solenoid Coil Monitor (SCM), the crest reached in e), above, was never more than a few percent higher than the coil's nominal current rating.
If you need additional info, let me know.
Regards, Phil Corso, PE (firstname.lastname@example.org)