Red Cascade is a red climbing mini. I grew one when I lived in the south, and the color is outstanding...a very deep blood red. The flowers are about an inch across, but there are gazillions of them. It's much prettier than the pics online. No fragrance though.

That Red Cascade is beautiful! Any thoughts on a yellow rose or companion plants for the fall/winter?

I'm sorry, I have no experience with yellow roses. However, if you aren't picky about color Brightside Cream is a super healthy climber with the most luscious smelling cream-colored blooms you could imagine. I got a huge flush in the spring and scattered bloom all summer. Once the trees leafed out she was in quite a bit of shade, so I suspect she's have bloomed much better with more sun.

Are you looking for vines to compensate for the dormant roses or shrubs to go under them or am I misunderstanding the request?

Because you askesd for continual bloom, one rose class that bears medium- to -large flowers, blooms more often than all other rose classes: The Tea class. Tea is an Old Garden Rose class, not to be confused with Hybrid Tea.
Tea rosebushes produce blooms over the entire leaf canopy, and bloom on average 25% more often than H.T.s. in a Mediterranean climate. Tea rosebushes thrive in warm climates and most are not cold hardy below zone 7, so you're in luck if you fall in love with the Old Garden Rose class, of Tea, as I have.
I suggest after you choose a few roses for beauty, post here and ask about how healthy they'd be in your part of Missouri.
I've listed roses that are presently available.
from rosesunlimitedownroot.com
'Adam' a beautiful Old Garden Tea of lovely pink and peach hues. I love the nuanced hues of 'Adam' and its' scent.
'Climbing 'Lady Hillingdon', its' creamy- peach and apricot hues are lovely against dark green leaves.
climbing Polyantha:
cl. Pinkie' is said to be a favorite in the South. It has earned the Earthkind designation in Texas for its' good health.
Hybrid Musk 'Renae' a very special rose, pink with lilac hues, and magnificent foliage. Scent is sweet and carries on the air in the right conditions.
These own root roses are available from Ashdownroses.com
I've grown the last 3, and love them all.
'Mme. Alfred Carriere' an Old Garden Tea-Noisette, very lovely white roses with a heart of pink to peach, one of my very favorite roses.
'Reve d'Or' honey and orange blended in one hue. Large rounded blossoms, ours is now starting to bloom, c. 1 month before most H.T.s begin blooming.
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on rootstock from hortico.com
'Alister Stella Gray' Noisette. a marvelous rose with small pom-pom shaped, yolk-yellow roses. Our neighbor chose this rose, alone, to be her wedding day hair adornment.

Best wishes for your having thousands of beautiful roses,
Luxrosa.