I was going to suggest roses, since I love the look of roses along a fence, but if you don't have at least 6 hours of sun, they may not do well.

Someone in my subdivision had a long fence put in and had some hollys with azaleas with a tree every 4-5 feet for vertical interest, with daylillies in the front which looked nice. I guess it depends on how deep the beds are along the fence and how much room you have.

Are you looking for shrub ideas since you already have perennials?

Shrubs that can tolerate wet/moist soil and shade:

Virginia sweetspire (Itea virginica)- several nice cultivars available
Florida anise (illicium floridanum) - an outstanding evergreen and underused native shrub
Leucothoe (Agarista populifolia) - also evergreen and underused (and native)

I'm open to anything - shrub, , shrub that can be pruned to a tree shape, perennial, vine - that will complement a picket fence. I'm even open to something planted in a large container to get around the damp soil problem in the shady border.

I hadn't thought about it before, but even planters hung on the fence is a possibility. Really, I'm open to any and all suggestions.

I guess it's too late to give you this advice but for anyone else maybe it will be helpful...
There are few things in this world as charming as a picket fence but...... if you don't make sure the pickets are not spaced too far apart, wild rabbits will come into your garden and destroy everything within reach and then poop it out all over the place. And they will do this slowly and cloaked by the darkness of night so in the morning when you come to find things looking not quite right, you sorta think your mind is playing tricks on you. Then after a little while the rabbits will tell other rabbits and they will claim your garden all for themselves.

Space your pickets close together and keep in mind the will shrink and the spaces in between will get bigger. Learn from my mistake so you don't waste your afternoon stringing expensive steel wire in the spaces and cleaning up the mess left behind. They will do a number on your lawn too if you have cool season grass.

Thanks for the advice. You are right that it's too late for me, but the fence only separates the front yard from the back yard. And the herb garden is on the front side of the fence, so little rabbits have easy access anyway. And oh do they love parsley and garden phlox.

My back yard is also protected by two vicious, enormous, super-hero, primitive hunting, wild fang-toothed dogs from the fires of the underworld - IN THEIR DREAMS! It's protected by a Corgi princess and a mixed breed clown. Fortunately, the rabbits don't know that and don't try to get through the fence.

Still need suggestions for what to plant next to the gate.

how about climbing hydrangea or gardenia by the gate

"My back yard is also protected by two [dogs]"
This was my biggest reason for keeping the rabbits out. I wasted way too much time cleaning up as many pellets as I could find today. They were scattered throughout the yard. I collected a few lbs of them! They are a canine delicacy. Unfortunately they can potentially make them sick too. I finsihed sealing off the back yard with wire today and hopefully they can not get in now. I have ES hydrangeas next to one of my gates. Oakleaf hydrangeas look nicer right now.. holds leaves much longer and has good fall color but blooms not as colorful.

I hadn't thought about hydrangeas. That sounds good. I've been thinking about a Penny Mac hydrangea. Does anyone have experience with them? I hear they are repeat bloomers - spring, again in summer, again in fall. Or maybe a Lady in Red. The burgundy leaves would be gorgeous next to the blooming anemones in the fall. Anyone growing Lady in Red?
Barbara, what can you tell me about climbing hydrangea? The only thing I have ever heard about them is that they are slow growers.