Look for an agency serving the immigrant population or getting the homeless into their first apartments.
We have several organizations like that. One serves immigrants, especially from Latin America, another serves a heavy Brazilian population. They have a standard food pantry, but they also have a household pantry, with clothing, toys, housewares, even furniture. Approved families (meaning those who are definitely destitute) can come in and "shop" for free.
Call your local coalition on the homeless too - see what they have or where they can refer you for people transitioning into housing.
A great resource, actually, is you local Congressman or state rep/Senator - they are usually aware of who's in your district who is reputable as they are also providing constituent services.
I think the big organizations that you describe are easier for a lot of donors - they just take everything and make it available to anyone. I have a problem with some of them, not others. Depends on whether they serve everyone, or if they discriminate (e.g. I don't give to the Salvation Army because their services aren't open to all).
If you are particularly trying to serve children, try Big Brothers/Big Sisters. Around here, they do regular pickups at your house, give you a receipt, and while much of the stuff is going into their stores, their target population is children of various ages.
You could also check on services for battered women and their children - they often leave their homes with nothing. If there is a local organization serving them, they can tell you if they need toys. They won't give you the address of a shelter, obviously, because those locations must remain secret. But I have met their volunteers in a central location and loaded my stuff into their cars. If you can't find an organization directly, call your local police department and ask for an officer who deals with the abused women. Tell them it's not an emergency but you are looking to connect with an organization that may need donations.