This is not a fully developed answer, so if someone could edit if they have any additional info.
Shir haKavod (Anim Zemirot) is typically recited in connection with the Shir Shel Yom (henceforward: shash"y). Shash"y is relatively mobile, for example it is Minhag Anglia to recite it* before Pesukei D'zimrah (source: friends from the UK). However, in Eastern Germany, France, and Chassidic and Sephardi communities, it is said right before K'riat haTorah (sources: davening in French and Eastern German minhag schuls as well as siddurim). Finally, the practice in the Rheinland, Eastern Ashkenazi siddurim, and Israeli siddurim is to say shash"y after Aleinu (this practice is consistent throughout the week) (sources: Rödelheim siddurim and machzorim, Artscroll, Birnbaum, Koren, and Rinat Yisrael).
As most siddurim print Anim Zemirot either right before- or right after shash"y, it is said in connection with the particular tehillah.
*The positioning given for Shir Shel Yom is based on when it is said on Shabbat an yom tov.