However in this case, whilst checking to find a station near Fleet, such as
Basingstoke, that might have 'open tickets', when comparing the price from
Fleet I now find the NR OJP shows Fleet (36 miles) to Waterloo has:
Super Off-Peak Day Return - £13.20
Off-Peak Day Return - £16.20
Off Peak Return - £24.20
The latter is the same price as the Anytime Day Return.
Has Philip just not found the ticket on the machine after all, or not
realised that the 'Off-Peak Return' is valid for a month?

I haven't found it on the machine, I do plan to spend some time
checking it out. *When buying "period returns" from the TO I am fairly
sure I have been sold two singles so I didn't realise it existed.
However I have checked on NRE for going out today and back tomorrow,
and as you say, an Off Peak Return to Waterloo is priced at £24.20
which is cheaper than two singles £25.40).

Well, this is annoying.

I can only think that earlier occasions included travel at least one
way at peak time - so two singles was indeed the only option.

I now know that I could have bought an Off Peak Return for travel Sat
AM returning Mon PM, and presumably could have got a Network Card
discount thus paying about £18 in total including two Oyster returns.
Or if I'd heard of a return to London International, an Off Peak
Return to London International is £24 which means with Network Card
reduction I could have got to St Pancras for £15/£16 including Tube.

It would help if an Off Peak Return was called an Off Peak Period
Return or an Off Peak 1-Month Return. Then you'd know what they did.

And it's another reason why SWT should keep ticket offices open and
not rely on machines: the staff can advise you of the best fare.