As you have a ticket issued by a train company you can do a changeover and pay the extra cost pro-rata. You can do this at any staffed National Rail station, though it will usually be faster to do it at one of the same company you bought the ticket from originally. Using London Overground is not advised, as their staff do not generally wish to do any National Rail tickets and will say “we’re TfL not National Rail”.
Whoever told you that you would need to refund the ticket and buy a new one was wrong. If it was a member of train company staff, that’s disappointing though not surprising. Some do not wish to perform this type of transaction as it is a bit of a faff. And others are not well trained. I have found the SWR booking office at Waterloo very poor myself, including deliberately selling a ticket different to the one that I had asked for (from Wimbledon instead of from Boundary Zone 3), refusing to sell a ticket to Birmingham Stations (“you have to choose which station you want... this is my ticket window and we’re going to do it my way”), and deliberately mishearing Sunbury, which SWR serves, as Sudbury, which it does not.