Drove car with radiator cap loose :(

Okay so I left the cap loose and I drove it for
about 15 miles as I'm getting off the freeway
I notice there's white smoke coming out of
he hood and the temperature meter was
going up. So I stopped as soon as I could. I
open the hood and notice the cap is loose
but there was still coolent in the radiator and
I put the cap on tight. There wasn't water in
the reserve and there was a 711 like 3
blocks up so I let the car cool down for
about 20 minutes and drove it to he 711 and
it was till marking it was over heating so I
opened the hood and let it cool off again for
like 10 minutes and the pored water in the
reserve. After I pored little by little water on
the radiator to cool it down (idk if this is
where I messed up) and waited another 20
minutes drove back home witch was like 5
minutes from there the temperature went
down but when I got home it was rising a
little so after that I did not moved the car. In
he morning I checked the radiator and it had
a little coolent so I filled it up. So I drove it
for a test run and I was checking the
temperature meter all the time and it wasn't
over heating but after I drove it for like 20
minutes I noticed that a little bit of white
smoke started coming out of the hood after
I would make stops or at a red light. Can any
one tell me if I just messed up my car?? Oh
and the cooling fan doesn't work so I don't
know if that might have something to do
with it?