It is definitely been my reading of it in the past, and I have treated passengers in the position of wanting to split their journey after departure when they had passed an opportunity to buy as travelling without a ticket before now - without any comeback.
Dave's detailed explanation is completely accurate of course. We deal with this at least on a weekly basis and frequently more often than that.
The usual arguments put forward at a barrier or on-train check during the final portion of the journey are 'I wanted to split to save money', or far more frequently 'I got a call on my mobile and because (
insert reason here) I had to stay on the train'
If there is evidence on the ticket held that the traveller who 'wanted to split' had purchased the first portion of the journey at a ticket office, or online, TOD etc. then an opportunity to buy both tickets has not been taken up by that passenger and a report may be filed.
A charge of RoRA [1889] S.5.3.b may be considered in these circumstances and in my experience, where the traveller has decided to challenge this and a summons has been issued, following legal advice some travellers have chosen to seek out-of-Court disposal and in all other cases that I have knowledge of, the Magistrates have been persuaded by the evidence and have convicted. None have been appealed.
If the traveller has started their journey at a station without facilities, or where only a limited ticket availability from TVM exists, provided a ticket has been purchased I always instruct inspectors to exercise discretion, which will then be informed by the travellers words & actions.
A traveller's wish to purchase a different type of ticket, but not holding ANY ticket when facilities were available to them before being checked, does not negate the liability to hold a ticket under National Railways Byelaws (2005)
When, as does occasionally happen, a traveller may have bought a ticket from A to B and gets a call on route, which requires that traveller to continue to C, if the traveller knows that there is a conductor/guard/inspector on the train and if s/he contacts that person before reaching B, declares the need to travel further the cheapest fare available ought to be charged for the remainder of their journey.
In other circumstances, as Dave has made perfectly clear, the traveller should alight at the point to which s/he has paid and re-book.