There are several hanging opportunities but not much in the way of backpacking in the general area of Melbourne Beach. If you feel like driving about an hour inland there's Tosohatchee WMA. They have a nice loop trail with a backcountry campsite. There's also a huge group camping area with nice spreading oaks.
A super short and easy hike might be Hal Scott Preserve. The hiking is basically on old roadbeds. There's also some good hanging down around Sebastian Inlet but I've never been that way. I believe Sunny's Trike lives down in that area so he can probably chime in.
Good luck. Looks like we'll have some great weather.