It still has to go through the Assembly as well. Despite all the "drama" from the Senate Democrats, the Senate is actually the "easier" house for CCW right now, IIRC CCW passed the Senate with a few more votes than needed for an override proof majority, so we could afford to lose 1-2 senate votes and still override the Governor.
In the Assembly, CCW passed, but two votes shy of a 2/3rds veto override majority, but two Republican Assemblymen who voted against the bill said they'd not stand in the way of the majority and would vote to override the Governor, so there it's critical we lose not one vote there.
Hopefully Ohio's, Minnesota's, Missouri's, and New Mexico's recent passage of CCW is adding some momentum, and provides our legislators with proof that this is passing most everywhere.