What is the best position to be in after the angiogram?

Lying down for 6 h. If it is done via an artery in the groin (femoral artery), then lying down at bed rest for 6 hours after is the safest thing to do.
Flat. After your angio, lay flat until the nurses tell you you're allowed to flex at the hip. You don't want to deal with the complications that can come from blood vessel disruption and it takes a short while for the vessel to reseal. Some mds use a "sealing device" to provide immediate closure in which case minimal rest is necessary but procedure costs increase.