I walked the Thames Path last year, and there is little choice in the Streatley/Goring area.
Just upstream is Cleve Lock with a café - not a recognised site, but you could try asking.
A couple a miles downstream there is a nice wooded section - so possible. On the way you will pass two pillboxes which might be usable as a bothy. There are many pillboxes along the path - some accessible others not.
I got ripped of at the YHA - as the hostel was full of contractors they charged me for double occupancy of the remaining room. A B&B in Streatley would have been marginally cheaper, but they were away that night. (It's run by a trail walker.)