Downriver from Hammersmith Bridge is Putney Bridge. The bridge is the nominal start of the annual Boat Race, the one in which Oxford and Cambridge always get through to the final.
There are churches at both ends; St Mary's, on the south end, was where the Putney Debates took place in 1648 to determine the shape of England's new republic, following the deposition of Charles I. You can debate what should be the fate of the monarchy there now too, in its coffee shop. The church is part of the Taking Liberties exhibition bike tour, featured in a previous post.
Continuing from Hammersmith Bridge, you have a few hundred yards of streets and lanes on either side which take you to the access stairs of Fulham Railway Bridge.