Opportunity is missed by most people because it is dressed in overalls and looks like work. - Thomas Edison
you certainly can do it in 1 pull. it depends on the mesh, ink, squeegee, off contact and the phase of the moon. But if it looks good with 1, roll with it.
Simple answer, no. solid isn't needed. In fact, sophisticated underbases use halftones and vary percentage tints depending on what color overlay is being used where.
Worse case scenario you hit it twice wet on wet, never stroke flash stroke flash. Usually we print it one stroke only and that is more than good enough. Your underbase is going to decide how smooth your final print will be, if it is not a smooth clean underbase all the top colors or even a second white will be rough if the first white was flashed. Once a ink coat is flashed its roughness is permanent anything on top will also be rough.What I have to ask tho is why didn't you experiment when you had it all set up? Use some scratch shirts and try different techniques and choose the best option. Also do not be fooled by the white base after the first print, sometimes the white needs to be printed a few times to get a really smooth and creamy print.