from my experience with focusing on the senses I already have the mere availability of the data is not sufficient. You really need to process it. The glass intervention works well because it also takes away the primary way of interacting with the world. If you only add sense most of it can be pretty much ignored as it doesn't bring any compelling extra value except for being cool for a while. Color TV was kinda nice improvement but not many are jumping on the 3D bandwagon.

So if you really want to go three eyed it could be a good bet it could be good from sense development perspective to go only new-eye mono for a while. Another one would be have a environment where the new capabilities are difference makingly handy. I could imagine that fixing and taking apart computers could benefit from that kind of sensing. You could also purposefully make a multilayered desk so that simply looking what is on the desk would require hand movement but many more documents could be open at any time.

Your brain is already mostly filtering out the massive amount of input it takes, making it quite expensive to make it bother paying attention to yet another sense-datum The sense would also require their own "drivers". I could imagine that managing a moveable eye would be more laboursome than eye focus micro. Having a fixed separation of your viewpoints makes the calculations easy routine. That would have to be expanded into a more general approach for variable separation. There is a camera trick where you change your zoom while simultanously moving the camera in the forward backward dimension keeping the size of the primary target fixed but stretching the perspective. Big variance to the viewpoint separation would induce similar effects. I could imagine how it could be nausea-inducing instead of only cool. Increased mental labour and confusion atleast on the short-term would press against adopting a more expanded sensory experience. Therefore if such transition is wanted it is important to bring the tempting good sides concrete in the practical experience.