Hitch-hike - -
I've hitched across a number of rivers including the Mississippi.
50 years ago there used to be numerous ferries - which are MUCH more pleasant for cyclists.
Now, the only ferries on the lower Mississippi north of New Orleans are at Hickman, KY and St. Francisville, LA - and the latter is getting replaced by a bridge next year.
Hitching is easiest if you pick a location with landings on both sides of the river - thus old ferry locations are ideal if the landings remain public launch points. Take a book - read for a while. My experience has been that the first people coming down to put their boat die laughing - then say, "Come on - - we'll take you across."