What about evergreen clematis? Before you do anything, test the soil around your arbor, consider how much sun you get there and how well the soil drains.
Were the arbor posts set into concrete? Might be causing a soil effect that disturbed the jasmine.

Maybe a variety so you have bloomers at different times of the year. If you want to add some annuals how about a moonflower vine/Ipomoea alba. They open in the late afternoon till next morning, smell heavenly and glow in the evening light, plus it's a very fast grower. They're a cousin to the morning glory but not as invasive. I've never had one come up as a volunteer. You can watch their big blooms open in the evening and the hummingbird moths flock to them. It's a nice fill in vine while your perennials catch up too.

At one time I had moonflowers that were blooming at night and cardinal climber during the day on the same arbor. Made for a pretty mix and the hummingbirds loved it. On cool cloudy days they'd over lap their bloom time in the early morning and it was pretty with red and white.

ginger, you're not pushing the zone, so you should check for other reasons why the jessamine isn't happy like dottie suggested. some of the older leaves will turn brownish as the new growth comes out in spring (now, here). that's just part of its natural cycle. but it sounds like you've got something else going on. most anything will take a year or 2 before it goes nuts. general rule is first year it sleeps, second creeps & third leaps. jessamine will cover your arbor in a year or 2, and you'll be trimming thereafter. anything that covers it more quickly means lots more trimming to keep it under control every year, so bear that in mine. you could grow some annual vines for a few years to help fill it out in summer. thunbergia, hyacinth vine, morning glories & the like will make short work of a trellis.

Not to sound like a broken record with my recommendation of clematis but why not venture out of the norm and use something other than an evergreen clematis like clematis armandii. In this area, the evergreen clematis are very often zapped early by unusual frosts like we have just had and the plants can get browned out leaves if the plant is not sited correctly and gets too much winter wind.

Why not investigate one of the clematis montanas? Depending on the variety, they can grow to a variety of lengths, have magnificent displays of blooms in an array of colors, and have relatively few pests if any!

See the link below and when you click on it, push Search Clematis at the top center. On the page you are directed to, enter montana in the "clematis name includes" box and then push enter. You will get a list of clematis with montana in the names and all the names with + beside them have pictures in their listing.

Here is a link that might be useful: Clematis on the Web

Nckvilledudes, it's good to hear that Clematis montana can do well here. The pictures I have seen of it look so tempting!

Thanks everyone for the suggestions. This will just be the 2nd year for our jasmine. Not all of the leaves are turning brown, so maybe it is just part of the natural cycle. The posts are set in cement though, so I guess we will just watch it this spring and summer and see what happens. I'll check out the clematis ideas.
I have planted moonvines several times and have not had much luck with them, they just don't seem to grow. I do like the idea of either cardinal climber and moonvine or morning glories and moonvine. I might give the moonvine another try.

ggwrn, if I baby it (moonvine) it'll die for sure. I just soak it overnight and then stick it in the ground where I want it once the soil is good and warm. Then I stand back and do my best to ignore it. It doesn't like a cold wet combination at all when sprouting. Good luck!

that goes for all the morn glory clan- wait til the soil is nice & warm, presoak the seed til it splits, then sow it where you want it to go. success! good luck.