Word for something which is “on the side” or which manifests “at the edge”?

liminal: 1. of or at the limen, or threshold 2. at a boundary or transitional point between two conditions.

The word liminal can refer to any threshold, including a physical one. The doorjamb between your home and the outside is liminal, for example (that's the ultimate origin of the word, limen), as is the seashore or coast (though that has a more specialized term, littoral).

A liminal smile can happen at any time at all, for any reason. It's just perceptible. Either visually, to the person seeing it, because it's subtle, not full-blown, just at the edge of the mouth; it has started there, as all smiles do, but progressed no further, just as you describe in your sentence. Or mentally, to the one doing the smiling, who feels the first feather of joy, or humor, or snarky condescension, in their hearts. It's a subtle signal betraying a subtle emotion.

An example can be found in a story in the anthology The Oxford Book of American Short Stories: "Susan Certaine ... is giving him a look he hasn't seen before. The liminal smile, the coy arch of the eyebrows ..."

Full credit goes to Dan Bron for providing the answer in the form of comments.

Answer derived from his comments based on SE policy here.