Should I continue putting suet out in a suet cage during the winter?
There will still be birds looking for food during the winter months. Just make sure the suet doesn't get too wet or freeze. You should be able to feed the birds during the winter months.
Suet has really always been the bird food for the winter as summer heat can really make a mess with suet feeders. I live in north/central Florida and the heat in summer is too much so winter is the time my birds get suet.
There are birds that do not migrate south in most parts of the US but it may take putting suet out for months or even a couple of years for those birds to find your food. Once a bird finds a food source, they will usually return to that spot often so be sure that once you start with food be sure to keep it supplied or the birds will give up and go elsewhere.
This is a page about making homemade suet. You can attract many birds by putting out suet cakes. You can easily make suet cakes at home and vary them to your local bird species.