Fields of the Wood, aka Climbing Rhode Island Red aka Climbing Red Kocher is also a lovely fragrant Red. You might also look into Red Fountain and Illusion. I adore Florentina, but I'm not sure it would get quite that high. If you don't mind a smaller flower look into Cl. Orange Triumph (it's red not orange!) or for Teeny flowers the charming Red Cascade. Of course, there's always Cadenza and Orfeo. I personally wouldn't consider Joseph's Coat a red. The issue is how much rebloom and fragrance you desire and which will hold up to your intense sun. They should all be hardy enough. Lady in Red is a very intense red veering to the oranger side, while Cl. Crimson Glory and EdH are more subtle blue reds and extremely fragrant. If you got Cl.CG you could plant the bush form at her feet. I'll have to think a bit more, but I grow most of these.
If you're looking for a once-bloomer Chevy Chase is gorgeous with a long period of bloom. If you decide on a taller climber with sparse rebloom or once-bloomer you could always plant a shorter one or one with smaller blooms that repeats well to intertwine, hide the bare legs and fill out the lankier climber. Here I'm thinking of something like the above mentioned Dortmund, Crimson Sky or Love Knot. You could even use another colour in this case. I think you've been given a great start!
