Under eye hollows need filler! What is the best product to fill the outer part under the eye above the cheek? (Photos)

Yes, I would absolutely agree that you need to place Restylane in the outer eye area.  In fact, without that area being corrected and only doing the nose area it would look unbalanced.  Please find someone in the area with lots of experience in the eyes and you find that it will look much better.  Also, you can consider Restylane Lyft since your hollows are deeper.

Dear fightingthehandsoftime:
I agree that this area should be treated. There are many different and appropriate fillers that your dermatologist or plastic surgeon will consider. One of the Restylane products is often used, although Restylane can result in temporary swelling, a Tyndall effect, or even lumpiness. For that reason, I prefer the fillers with a lower cohesively, such as Belotero or the newer Volbella. The most important thing is consulting an expert injector, as advanced training and experience is required for a good result.
All the best,   Dr. Clark