For cars of this vintage, there were two different coolant overflow designs in place.
One design used a simple system, with an overflow bottle. There was a single hose that ran from the top of the radiator to a catch bottle. For this system the radiator cap has a fancy double seal system. One seal holds pressure to 13psi. Above that anything else goes into the catch / overflow bottle. The other seal in that cap uses a very light spring. This seal functions after your car is shut off about 8 hours or so, and has time to cool down. This seal allows coolant to return from the overflow bottle back into the radiator.
The other design uses a coolant bottle as in process reservoir. Generally this bottle is mounted high in the engine compartment. It will have a high pressure sealing cap. Generally in this system there will not be a cap on the radiator. In this system coolant flows thru the bottle all the time. Its pretty easy to see the coolant level, and this design precludes air bubbles from forming inside the engine from poor fill procedure.
I'm not sure which system is in your car. Note: One thing that would help here is for you to edit your question to add a photo of your engine compartment and a photo of the hose you are looking at. Possible that you can get access to a cell phone with a camera?
Recommendation: Purchase a $30 Chilton or Haynes Paper Repair Manual. From your description you've easily spend more than that on parts you probably didn't need. That manual will show you in pictures what parts go where. There shouldn't be an unattached hose for your 2004 coolant system. (A 1960 system, yes, but not 2004)
I don't understand your comment about
put blocked seal permanent head gasket fix.
What keeps a system cool is good airflow thru the radiator, and a completely filled coolant system with the proper hoses and pressures. I'd check that the coolant pressure cap is functioning correctly. (The cap maintains a 13 psi pressure inside the system. @ 13psi water boils around 241 degrees F or so... and this helps the car engine stay cooler.)
The service manual will tell you how to test the coolant pressure cap, and how to test that the cooling fan is working correctly. It will also tell you how to fill the coolant system to bleed off the internal air pockets in the head of the engine. Its not as simple as opening up the fill port and pouring in distilled water. Oops, you are using 50/50 antifreeze an distilled water aren't you? If not, that would be bad.
There are a whole lot of questions in this group on generally good coolant practices. Do a search for