The Tiferess Yisrael on this Mishnah asks exactly the same question.
He answers that the Gemara in Bava Metzia 25a establishes that the Mishnah is talking about young pigeons which can hop, and the Gemara in Bava Basra 23b teaches that such a pigeon does not hop more than fifty amos from its dovecote, but in the previous amud it teaches that in fields and vineyards it hops even more than fifty amos from its dovecote.
Therefore, if the Mishnah had taught the phrase "tied birds" after the phrase "or on pathways in the fields", I might have thought that since the Mishnah does not make a distinction whether the birds were found inside or outside fifty amos of neighboring dovecotes, it could also be discussing a case where they are found outside of fifty amos, and thus I might have thought that only when they are tied do we have to be concerned that someone placed them there and so he is not allowed to take it. But if they are not tied then even if he finds them more than fifty amos from the dovecotes he would be allowed to keep them since they may have hopped from one of the dovecotes.
Thus, I might have thought that the phrase "tied birds" only referred to the the case where he found the birds on the pathways in the fields, but if he finds them behind a fence or behind a wall outside of fifty amos then even if they are not tied he is not allowed to take them since they do not hop more than fifty amos from their dovecote. This is what I might have thought.
But in fact all this is not true, because the Shach in Choshen Mishpat Siman 260, note 28, proves that our Mishnah is speaking specifically about finding birds within fifty amos of the dovecotes. Therefore, the Mishnah deliberately puts the phrase "tied birds" after the first two cases, to teach us that even in these cases it is talking about finding the birds within fifty amos, and that only when they are tied does he have to be concerned that maybe someone placed them there.