It was in storage all winter? How many miles have you put on it since you took it out of storage?
Its not abnormal if the car is sitting to have it smoke when you first fire it up, its usually the result of oil seeping from the top end through the lifters & onto the pistons. Harmless really.
If it keeps smoking its likely to be stuck rings, a common problem in Celica motors & it would be aggrevated by storing it. Despite all kinds different products, I never did get the rings unstuck. The rings get a gummed up & stick inside their grooves on the piston & fail to seal the cylinder walls.
Another possibility may be the vacuum going from the top end of your motor to your Intake
, it could be sucking in oil.