Slightly tricky. From 1/2CL S rho Vsquared, if you maintain IAS in level flight and S does not change then CL doesn't change either. Fowler flaps increase S, but this does not increase CL for level flight. Strictly, with Fowler flaps CL required for level flight goes down.
CL max available increases with all types of flap - after all, flap is called a "high lift device". So will drag. The pilot adjusts attitude and power to maintain speed and S&L