TT points to the sequence of dressing Aharon and the comparison with the command in Shemos 29 (verified via Essential Torah Temimah).
ויחגר אותו באבנט — ובציווי (שמות כט:ט) כתיב וחגרת אותם באבנט. הא כיצד? אהרן קודם לבניו. וחגרת אותם — ללמד שהאבנט של כהן גדול ושל כהן הדיוט שוין (שניהם של כלאים).
“And he girded him with the girdle” — and in the command (Exodus 29:9) it is written “and you shall gird them with the girdle.” How is this reconciled? Aharon is clothed before his sons. “And you shall gird them” teaches that the girdle of the High Priest and that of the ordinary priest are the same (both are of kilayim).
Custom — reconciliation of narrative order with the command, and equality of the girdles.
Aharon is fully clothed before his sons begin to be dressed. The girdle of the High Priest and that of the ordinary Kohen are the same (both of kilayim).