Don Juan is my hands down favorite. It's done the best for me, blooming 10-11 month out of the year and growing like crazy: I live near Houston.
Don Juan is also disease resistant. I don't spray it, fertilize it, water it, anything. Easier than an old tree.

Lady Banks is good for Austin, but only blooms once a year.

I don't have a lot of climbers, but have been very impressed with Westerland so far.

i like my climbing souvenir de la malmason (don't know if spelled that right) a lot also the rev d or and mutiblis

the sdlm only blooms profusely in spring and sporadically after that but in spring it smells marvelous the best
(it's pink)

I second Rev D'Or. It is blooming even in this heat. Very small blooms, bur blooming.
Mine grows as a rambler. I did not realize it could be used as a climber when I planted it.

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Plantmaven,
Your roses are beautiful. It's amazing that it will bloom so well in this heat. Do you fertilize and water all the time?

Red cascade a mini climber. It is nearly evergreen and ever blooming.

I apologize. I should have stated that was a spring picture. The ones now are not as many and about 1/4 that size.

They are sometimes fed if we remember to do so.
Today I bought Black Kow composted manure. A friend swears by it. She said because it is composted, it does not burn them. So I thought I would try it.