If you mean you're planting Jasminum officinale, that spreads sideways over time as well as up, so the spread will take up probably 10-12 feet from ground level upwards - there isn't really space to plant anything else in that narrow strip in front of the fence. Even if you chose something better behaved such as climbing roses, or perhaps Clematis cultivars, there still isn't space for larger shrubs like Bay or Photinia - both will grow to small tree proportions. Is the deck raised above the ground, so there's a space between the ground and the underside? Exactly how wide is the strip of soil between the fence and the deck?

You should re-think this whole planting idea. Honestly, I don't see much space for anything. Jasmine requires full sun to produce its intoxicating fragrance. That's 6 hours of full on noon day sun "think South Carolina". That tall hedge behind the fence is hogging all of your sun. Jasmine won't do well. I would keep that space open, clean & passable with maybe a flagstone path & gravel mulch. Hanging baskets?

My first reaction was that there isn't really room to plant anything that will do what you are after. The hedge and fence are quite attractive in themselves. You might consider a row of small neat evergreen shrubs? (eg only abut a foot tall)Looking ahead space should be left for fence maintenance with preservative or similar even if it's a neighbour's fence.
This website has several suggestions that you might like - all you really need is to soften the edge of the deck and cover the soil?
https://www.rhs.org.uk/advice/profile?pid=845

If you try to plant larger shrubs they would have to be cut back so fiercely that they would look unattractive for much of the time. Agree with Bamboo re the jasmine. Just the one you have will cover a big area on its own and you might need to consider stringing some horizontal wires between the posts so enable you to contain it somewhat.

I've got a fence covered in Jasmine but it's really bushy & leafy most of the year. It's definitely not neat.
The early Morning thing is an old wives's tale/red herring...
As I like to grow food I'd suggest maybe Fig Brown Turkey or a grape that's good in Northern climes but Wisteria can be spectacular

depending on the size of that area from floor to fence I would choose some plastic pots and have violas or maybe short wall flower for winter colour, bulbs for spring colour and then begonia semperflorens or trailing geraniums to take you through the summer months. If you do not want the pots to be visible from the patio attach patio flooring boards between the uprights of the pergola. I would forget the jasmine to climb your posts. It is a thug in my garden and I am trying to get rid of it. You would probably get away with winter flowering jasmine.

I think the best option here is to use a selection of decent pots along that narrow strip, the hedge is lovely so I would not hide it, but plant the pots with some decent plants, you could hide the fence completely and just let the foliage creep slightly above it, the effect will be fantastic, so many shrubs will do well in pots, and for good foliage and flowers then a selection of Hebe, all evergreen, in many shades, golds,greys,purples, etc, senecio does well in pots, too many to mention, large pots with Bamboo placed at various points on the deck works well.

Do you like ornamental grasses? You could plant the evergreen Bowles golden sedge alternating with Ophiopogon nigra: smart and easy!