To connect with other birders, have you tried the Birding Pal organization (http://www.birdingpal.org/pa.htm)?
I count at least 8 that are in the city or adjacent area.
Fairly large for a city park, Fairmount Park is adjacent to and on both sides of the Schuylkill River. I have played in the creek of Wissahickon Parks 60 years ago as a youngster but no telling what I was exposed to in those waters! There are many trails to explore there.
Tinicum has a good record of species found but the airport is right there so birding by ear may be intermittent. Not that it would matter terribly for waterfowl...
I assume you've tried "Birding Philadelphia" on the internet and eBird reports for May sightings.
Cape May has a fabulous record but you have a single day if I am reading your post correctly. Do you want to spend most of it driving?