Answer: Here are couple of suggestions: Campsis tagliabuana ?Madame Galen? (Trumpet Vine); I think it's one of the choicest summer blooming vines. This is a red form of the classic hummingbird magnet and it blooms all summer with all the fine attributes of the traditional radicans.
Clematis "Dutchess of Edinburgh" is a white flowering vine; Clematis 'Henryi' also has white flowers, if you'd like to match your climbing hydrangea.
For contrast to your hydrangea, consider Lonicera Sempervirens ?Blanche Sandman? (Coral Honeysuckle) with reddish-orange, trumpet-shaped flowers opening to yellow. the flowers first appear in May and continue to bloom off and on until frost on this very vigorous native vine. This Honeysuckle is aphid-resistant and an incredible hummingbird magnet. There's also ?John Clayton?, a yellow flowering honeysuckle.
Best wishes with your new climber!
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