In my ward growing up one of the scoutmasters was a non-member father of one of the boys in the troop. This was one of the nice side-benefits of the Church’s involvement in the Boy Scouts: it was the perfect calling for the non-member husband of an interfaith couple who wanted to be involved with his kids in the Church’s youth programs. I’ve also seen cases where the proverbial, completely non-believing, occasionally-comes-to-church-to-support-their-spouse member was wholeheartedly willing to accept home teaching assignments when we all knew full well it was more akin to a social worker arrangement than a ministering priest one. 

(One could argue that male ministering assignments should only be given to those that can do blessings and the like, but frankly in most wards the demand for ministers, people that check up on you, even non-religious ones, so far outstrips the supply there’s little reason to not accept help where it’s available). 

More generally speaking, it’s nice to have some sort of low-stress calling that allows part-members to be involved in their family’s faith community without the religious commitment. A lot of the formula would be the same; for example, I don’t see why they couldn’t be set apart for a calling (if they’re okay with that), similar to how you don’t have to be a member to receive a priesthood blessing. Of course, this would have to be handled gingerly; they don’t feel the same obligation to accept callings and it would typically have to be something that involved their family directly.( I guess if you could convince a non-member to run tithing down to the bank every Sunday more power to you, but I doubt that would work.) If the Church is already asking for their family’s time during the weekends and mid-week the ward leadership would have to really read that particular room well in order to know when a calling for them was appropriate. 

It presumably wouldn’t involve teaching, but even then with the constant struggle sometimes to find enough teachers I’ve more than once thought about a hypothetical situation where in an emergency we’d ask the ward, never-member atheist to draw from his or her general Western cultural knowledge to teach golden-rule ethics from the Bible to Sunbeams. 

So maybe I’m just being non-creative, but post-Scouting the only ready-made callings that I can think of that would work for something like that is maybe pianist or choir director, but there may be custom-built callings that could work in those very particular situations.