In this time of increasing aggression from the forces of Christianity in the USA, I think it's important to have a place for the non-religious to turn when they need services provided from a spiritually neutral perspective. Especially people who are in any kind of trouble; it's bad enough the situation they're going through, without having people try to exploit their vulnerability by proselytizing to them, and worst of all if you're paying them by the hour to do that.
Please help spread the word about the Secular Services Directory -- details at the link. The directory especially needs business listings, which are free to create, so if you know business owners with a secular outlook, please ask them to sign up.
Your religion is not the answer to my problem. Solving my problem is the answer to my problem.
I'm not familiar with these services in general, but it seems that even many services that are provided by non-secular orgs aren't heavy into converting anyone. Catholic Charities in particular (in my area) just want to do good for the community.
Meanwhile, recalling your request for suggestions for secular substance abuse recovery, I just (accidentally) found https://www.sobernation.com/special-resources-for-atheists-in-recov... has some interesting links, from 2012. All links there (but one) still work.
@Tyler, you are at the top of my google search, so good job there! But your search feature wouldn't produce any results for any of the org names or keywords I tried. I only mention that in case you've already entered some relevant orgs, yourself.