It sounds like a lovely idea, and I don't want to sound negative, but from my experience with lilac root systems, a clematis might find it difficult to get established if it is planted nearby.

Having said that, if you're looking for something that flowers after the lilac blooms, consider those that bloom later in the season on new wood (pruning type 3) of which there are a multitude to choose from. One of my personal favorites is Venosa Violacea.

Have fun choosing.

I've done this myself :-) Too many clematis and not enough support structures, so planted them on darn near any shrub or tree large enough to accommodate them!

My lilac tree (single trunked) is a bit larger than yours and at least 60 years old. Had no problem getting several clematis established at its base. I chose two type 1 clems (1 macropetala and 1 alpina) as I didn't want to worry about pruning needs and I wanted something that would bloom early, before the tree leafed out. A montana would work as well. I was also just a bit skeptical about how well a later blooming clem would show up in all that greenery once the tree was in full leaf. If you were to opt for a later blooming variety, I'd suggest a type 3 also. These tend to be larger vines than most type 2's so will extend well up into the lilac and pruning is optional - if you don't prune, the leggy portion of the vine is easily disguised along the trunk/branching of the lilac, with the topgrowth and flowers reaching up into the sunlight.

A point raised by gardengal is how many of the type IIIs would show up in the greenery of the lilac. That is easily solved by choosing one that is lighter in color such as Betty Corning, Alba Luxurians, Prince Charles, Maria Cornelia, Blue Angel etc. or by choosing a color that would contrast nicely with the green foliage like the strong but lighter color of the Duchess of Albany, which is a very vigorous grower in my garden. Another option would be one of the yellow flowering tanguticas. The dark purple colored clematis seem to blend in well with most green foliage plants so would steer clear of them. Venosa Violacea as pointed out by the first poster would work since the interior of the sepals are a pale lavender to almost white.

I have a Huldine growing up through a lilac. It bloomed prolifically in its second year and seems to be doing well. I did plant it a few feet away and used a garden stake to encourage it to grow across those few feet to the shrub. I also read that planting it on the side opposite of the brightest light would help it grow into the shrub and it seems to have worked well.