I go to random nursing homes .
Well , not now w covid , but for last 13 yrs . They always need volunteers .
Old folks are lonely . Nobody to talk to . Nobody to play a game . Gardens that nobody has time to take them to , nursing homes always understaffed . Even to stroll them out for few , it makes a difference to them .
Also , if u have dogs that are well behaving , nursing homes have pet therapy sessions , and visiting w your dog is the highlight for many of the residents .

I like helping my local Historical Society! And Planned Parenthood. And my local zoo...doing education for the public.

I’m seeing organizations in the comments that I didn’t know existed, and wanted to thank you for this post.

Here’s a link to a list of atheist vetted charities [thaumaturgical.com]

Public schools love volunteers

The Foundation Beyond Belief. They do good work around the globe, finances are transparent, no dogma. I support them.

[foundationbeyondbelief.org]

I work with Moms Demand Action and Planned Parenthood! North Texas food bank always needs goods too! Good luck and be safe!

One of the big things that sold me on Big Brothers / Big Sisters was their no religion rule. I'll strongly recommend them.

For example, years before I joined we had a Halloween party at my brother's ranch.
There were about 250 kids -- all in costume -- each with an adult companion (that had gone through a background check) -- and each ready to behave well.
BBBS is a very good group.

I support the ACLU

Most animal shelters. The people may differ by location. In Oak Park Illinois, the folks never talk religion. Down in Charlotte, my daughter volunteers with Saving Southern Kitties and they are very liberal due to the folks who started the shelter.

If you like kids, contact Child Services in your county. They train people to supervise court-ordered parental visits. Also food banks, shelters, Planned Parenthood (esp. to walk patients from cars into buildings), senior centers.
If you like to be active, Habitat for Humanity, walk dogs for an animal shelter, or foster a dog or cat.
I used to be a hospice volunteer and found it very rewarding.
Also, you don’t need an organization—on your town’s Facebook or Next Door page, ask whether there are elderly, disabled, or otherwise needy people who could use a hand with yard work, groceries, or whatever skills you can offer.

Yes, but I live in the UK where most of the charities are not religiously affiliated. It may be a lot tougher in the US to find such a thing, you have my sympathy, not being able to make the contribution you would like because you are excluded on the grounds of faith, must be hard.

Have you considered starting your own ? Especially if you could find two or three other people in your area who are in the same place, at the very least you could open a small charity shop/stall and raise some funds, which you could spend locally or pass on to a larger national/international charity.

I do help out at a Salvation Army store. Have been doing it for 15 years. Not once have anyone there suggested that I either join them or come to church, not once, but then this is in Australia.

Since 2006, I have been a volunteer college mentor at the local high school. I help students- children of immigrants- write essays to apply for colleges and scholarships.

One of my best success stories is Brenda, who won $269,455 in scholarships and grants in 2016. She is a medical doctor and pathologist.

Being a college mentor is the most rewarding volunteer work I have ever done.

I have never found one. I volunteer at the VFW and the Eagles, and used to volunteer at the Moose…and long ago the Jaycees. They all say the pledge with “under god” and pray at the beginning and end of meetings. I stand and am respectful but do not participate. And I omit “under god” when reciting the pledge. We have a local Freethinkers group, but they do no charitable work. I do know that both the Tucson and San Diego Atheist groups do volunteer in their communities, so that would likely be your best option if you are in a larger city.

I've volunteered with an animal shelter, with an after-school tutoring program, and with Citizens Climate Lobby, which is building support in Congress for a national bipartisan solution to climate change. The last group I discovered online while searching for volunteer groups.

Check with your local library to see if they need volunteer help, LGBTQ+ Office at a local university (if there is one near you), homeless shelters…especially for youths, UU “church”, animal shelters, food banks, etc

I donate platelets and plasma every 28 days at the Memorial Blood Center (MN and WI)
. . . alternative :
[redcrossblood.org]

I volunteer at The Food Group Warehouse Tuesday mornings.

I consider the Freedom From Religion Foundation [ffrf.org] a charity, and I support their work wholeheartedly. Probably not what you had in mind, though.

Your local humane society!

American Humanist Organization. [americanhumanist.org]

I seem to have started my own local chapter. You can too. More on that later.

[aaccsecularhumanists.org].

Delaware Reserve medical Corps

I did volunteer work for gay organizations. The Los Angeles Gay & Lesbian Center and Christopher Street West (Los Angeles Gay Pride).

If you are looking for something quasi religious. You might look at the Unitarian Church, which is nondenominational. They have some atheists who attend for the sense of community it provide.

Lions Club

how about the Ayn Rand Foundation?