Nope. A tampon can't travel beyond your vagina because your cervix blocks the way (while the cervix does have an opening, it's too small for a tampon to squeeze through). That said, that Tampax can move out of easy grabbing range.
So what do you do if you're searching but can't find the short string? Try this: While sitting on the toilet, bear down like you're having a bowel movement, and reach inside with your (clean) fingers. Feel around and up for the end of the tampon. If you come up empty, don't spend hours playing Where's Waldo. Instead, just call your gynecologist, and let her fish it out for you.
Too embarrassed to alert your doctor? Don't be; my ob/gyn friends assure me that they take on this search mission all the time. Also, it's very easy to forget whether you have one in or not, so don't be surprised if your doc finds more than one (or none at all)!