I, for one, buy it. Andy's hat is Emily's old hat. The lacing is the same, and you can see where the white band used to be. I also find no holes in the theory.
Toy Story was originally a stand-alone movie (as such movies usually are), but when they went to create a sequel, they incorporated elements from the original. This required going back to the original and studying it for things that they could tie-in. The original poster on Andy's wall is "ABC Roundup", but that became "Woody's Roundup", an old black-and-white TV show. ("ABC Roundup" is presumably a newer, color version that came back later, much like the Mickey Mouse Club and other kids shows do from time to time.) Anyway, Andy's old hat didn't look like Woody's, but it was a cowboy hat and they used it as the basis for Jessie's hat. Then someone got the idea of working it into the "When She Loved Me" sequence. Perfect fit, and a great Easter egg.