I can't see £4.40 or £2.60 in either NR fare table. It's got to be weird behaviour triggered by the same station exit. I've tried to send the poster to my site to see if he'll let me know the stations.
A Zone 1-3 peak through fare is £4.40. The only other fare for that price is a peak TfL Zone 1-5 journey. And the only fare for £2.60 matching at least one of those zones is a Zone 2 - 5 peak fare. I'll discount the possibility of this person using Zones 7-9 or having a railcard.
Hmmm. What's interesting (and not surprising) is the following;
LO - TfL station -> TfL Only fare
LO - NR station -> NR Only fare
Obviously with some exceptions for stations PAYG was accepted at pre January 2010. It doesn't appear to be possible to trigger a through fare using LO (I tried Dalston Junction to Charlton.)
The only certainty I can draw from this is that the origin or destination is a station in Zone 5 where a TfL only fare is charged. There aren't very many, especially if the person's account is assumed to be clearly written, that both tube and Overground serve. That would limit one of the stations to being Harrow and Wealdstone.
If that's correct then the only station on the other end I can think of which matches with the fares and distance quoted is Shepherd's Bush. Shepherd's Bush LU to Harrow can't really be done without changing somewhere in Zone 1. Shepherd's Bush to Harrow via Willesden Junction would be a Zone 2-5 fare.
If this is correct then the person has triggered an OSI at their origin, which is something I've looked at in my Oyster case study as there is a money saving potential but wouldn't do in practice as it's intentional fare evasion through exploiting a weakness in the programming.
We shall see.