Included in your cooling system is 3 basic areas. Two cool which is the radiator and the heater core. The other builds the heat, your engine.
If you have an water hose that goes directly into your engine, it's much easier. Some engines only have have water going in and out of the water pump. The y pipe I am describing was a home made gizmo I saw at a radiator shop. As said, it had water going in one side, air going into the other, and the outlet for the both of them you connect to an existing water inlet. while keeping the water flowing all the time, you periodically hit the air to get a higher pressure going in, loosening stuff up. The flush you used probably loosened up a lot of gunk and rust build up. Leave the bottom hose off the radiator and let it flow through. Move the water/air hose to different areas to flush as much as possible. You should have drain plugs in the side of your block, depending on the engine. If it's just gunk, it will clear up easy. If it's a lot of rust, it's more difficult because it will settle in every lo spot in the engine.
I would do the radiator first, and then the heater core, and then keep them out of the flush procedure. It takes time.
If it's a cast block with a block drain plug, you can get a nipple from the plumbing supply and use that as an area to flush through.