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You don't want the southside, it is either mud and dirt on the river path or a boring ride over the B349. Plus you miss the cool dark section after Barnes Bridge.
Just cross Barnes as usual and ride to Hammersmith Bridge.
Ride the river path towards Putney until it spits you out on either Woodlawn or Stevenage. At the end, turn left at the T-juction then right into Fulham Palace Road.
Over Putney, back immediately on the Fulham Railway Bridge and continue as before.
This looks to be the way.
Will do a reccy of the Hammersmith to Putney route for the group, Fulham Palace Road is the direct route but can be busy. Will see if the river path is a good surface one evening next week.