What you remember as a kid was probably what I call "Wild Roses". they probably have a name though I don't know it. The ones I usually saw growing wild were pink.
Basically, there are three kinds of roses: Hybrid teas, floribunda, and climbing roses.
For your split rail fence, the climbing roses or the floribunda would probably work best. From your description of what you are wanting a scrub like then, that sounds like the floribunda rose bushes.
If I were planning what you are wanting to do I would research which roses are best for your area. The ones that would be best for the part of SC you are living in.
We buy our roses from
HERE
You can request a catalog and spend hours looking at all the beautiful roses. The hundreds of different colors, too!
There will probably be local nurseries that sell roses and they would be a good choice, too.
Here is an article on floribunda roses:
https://www.thespruce.com/floribunda-rose-varieties-4140326
These look like the ones I remember seeing out in the country:
From:
https://www.thespruce.com/climbing-rose-gallery-4050807