Only you can really define what is productive to you, but people are always needed for running the shelters, working in the kitchens and more. It isn't glamorous at all.
People need to be transported from one facility to another, to medical appointments, to court, to counseling and more.
I currently work with youth at risk, helping them finish their high school degrees and families who are right at the brink of falling into serious problems.
I can't really tell you which organizations to volunteer, as there is a spectrum of them; some are religious like the Salvation Army and some have social and political agendas that may not align with you. Without recommending or condemning, for example, you will find organizations who work with pregnant teens who are anti-abortion. This may not be for you. The same goes for sexual and gender identity issues.
One thing that seems very unpolitical to me, and it is how I started, was volunteering at a local shelter. If you have more affinity with helping women, there are shelters that deal with them. If you have more affinity with dealing with people with addiction issues, there are shelters dealing with them.
Because of my background in the military, education and addiction, I find affinity working with veterans, addicts and at risk youth.
The problem is daunting, and the needle feels unmovable most of the time. In the big scheme of things I feel powerless, but I know reducing local misery is possible and is something I can help with.