Either one should work fine for your purposes

if the do the standard two passes, thats really not a lot of cores. maybe you could ask them if they would do twice as many passes because you are overseeding, for a small premium.

That's what I was thinking like horizontal then vertical

Where are you? Why not overseed in the fall?

I'm in St. Louis and the yard is looking pretty bad. I want to get something down now then again in the fall.

Don't spend a lot of money on seed for the spring. If your budget allows, use the good seed in the fall.

Cheap seed is basically weed grass to me. I'll only use better seed on my lawn no matter when I'm seeding. Use cheap seed if you don't really care about a quality lawn or can't afford it.