Can I put one brand of paint over a different brand of paint?
Technically yes as long as it is the same base (assuming your existing valspar paint was latex, you would need another latex paint).

The sheen or finish of your paint (flat/satin/semi/gloss) is of no concern and shouldn't provide a problem.

However, doing so likely won't do what you want.  Assuming your dissatisfaction in texture is more than simply the difference between flat/gloss paint, it means the surface you painted was likely the source of your roughness.  You need to smooth (sand) the surface before painting.