You could get some 14-16 gauge wire and eye screws. Spools of wire and little boxes of the eye screws are available at the big box stores. Run the wire through the screws. I'd drill small holes first so the wood doesn't split.

I bought a roll of wire fencing at Lowe's and put on mine...just cut what I needed and the rest for my future projects...Jeanne

Jeanne,
I hate to ask but..... how did you attach?
Lois

There is a thread called "Hardware Cloth for clematis and other vines" at page 5 (could move to 6 later) on this forum, which has a good discussion and several ideas to choose from.

The "Roll" can be cut to leave "prongs" on the sides which you can wrap around your existing Trellis..it's all in the cut of the wire...

Here is a link that might be useful: Wire fencing on trellis