Depending how your house is constructed, it may be possible to run a CAT V/VI cable through to the southeast corner, maybe your skirting boards have cable run slots in them or perhaps run up to the roof space and down again, then if you want WiFi access at the remote end, just use an access point device. A cable would be your fastest link solution.
If you can't get a cable through, Then Bob's suggestion is the next best. One note though, I've found that power line adaptors work best is they are on the same ring main. My house has two separate rings off the consumer unit, so the signal has to run back there and out again and I do get some interference on occasion.
If neither of these work for you, then the range extender is your option. As has been mentioned earlier, if you can get the same brand as the router inside your Verizon box, that will give you the best chance of success. And has also been suggested, the conventional location for the extender is midway between the router and the extended destination. However, there are situations where other locations are more effective. The most likely cause of your poor SE signal is something (a brick wall, a kitchen or some other electrical device) in the direct path from your router. Suppose you had say a 20 metre hallway in line of sight from your router and then a 5 metre hallway at right angles the SE corner. in this scenario, the best place for extender is in the apex of the corner where the two hallways meet.
Final thought, what waveband are you using? Some routers offer both the 2.4 GHz and 5 GHz bands, in which case the first choice might be the 5 GHz band, since that would be free of interference from microwave ovens and cordless phones etc. Except that the 5 GHz signal is notoriously bad at passing through brick walls and other obstructions. So in some cases, the lower 2,4 GHz band may be a better choice. I don't live in the USA, so I know nothing of Verizon routers but it might be worth investigating.
Hours of fun ahead!