Saving time finding a result by guessing first?
Yes, it absolutely would save time.  You are quite right, and you are onto a good idea -- that idea is frequently used to speed up image matching.

The core challenge is: how do we come up with an estimate where the result might be, more efficiently than using the correlation algorithm?  It's not obvious how to do that, short of doing a correlation (which wouldn't actually save any time).

Fortunately, there are ways to do that.  A standard approach is to use an image pyramid.  You downscale the image to a lower resolution, then look for a downscaled version of the T in that smaller image (again using correlation).  Because the image is smaller, the correlation calculation goes faster.  Then that gives you a pretty good estimate where the T is, and instead of searching the entire full-resolution image, you only need to check a small portion of the full-resolution image.  If you find multiple matches in the downscaled image, you can check all of them on the full-resolution image.