The TfL paper ticket of £4 is valid on London Overground services which are also a TfL service.

The confusion lies that London Overground is also a National Rail provider and Southern provide a service into London Bridge and Victoria, so dual tickets are available.

If the original poster purchased a Underground/DLR/Overground Z1-3 single from Oxford Circus for example without asking for HOP/Forest Hill. As long as the passenger travels via Whitechapel, Shadwell or Canada Water, the ticket is valid and if going to FH will open the barriers.