Do you have a favorite brand that also makes diapers? I really like the dish soap from Seventh Generation, and they also have baby diapers. Sign up for their “Generation Good” club and you might get selected to receive product samples! Start noticing the companies you use that also make diapers, and reach out to them to see if you can try a sample from them!
But honestly, cloth diapers, as I mentioned earlier, have the most robust free diaper programs and they're the easiest to access. Groups like Giving Diapers, Giving Hope and The Rebecca Foundation's Cloth Diaper Closet offer free cloth diapers to those in need. Alternatively, you can get free or very cheap cloth diapers from online forums like Diaper Swappers that specialize in recycled (highly laundered) cloth diapers. Change-Diapers.com is a great resource with a link every Friday for cloth diaper giveaways they've found online, too.
UPDATE: I recently found another great way to get free baby samples! Check out the Everyday Mom Sampling Club. When you sign-up (it’s free) you’ll receive a new box of samples every month as part of the program. You’ll get samples from companies like Gerber, Pampers, Huggies and Enfamil. I’m not sure how long this promotion is going to run so you should check it out ASAP. Sign up here.
Why talc free? There is some controversy regarding the safety of talcum powder. Back in the day, it contained asbestos (a known carcinogen), but from the 70's forward, all talc used in bath and body products in the US is asbestos free. So what's the issue? Some believe that even the asbestos free talc is carcinogenic and studies have been unable to prove otherwise.  Bits from the American' Cancer Society's page "Talcum Powder and Cancer" like this:

The charity type agencies listed below receive donations of diapers (or the money to pay for them) from local businesses, civic groups, community businesses, schools and faith-based organizations. Free samples can also be provided to some clients. The goal is to help ensure children and their parents have diapers, wipes, and other personal cleaning supplies to ensure that the child is healthy. Find a listing of resources by state below.
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