This kind of profession would take place after the entrance procession but before the Mass itself.
The candidate(s) for the third order would make the profession in front of the priest and be invested in the vesture of the third order (habit, medal etc). After the profession ritual is complete the celebrant vests for Mass at the sedilia or sacristy and Mass proceeds as usual (whether low, sung or solemn) with the candidates attending Mass for the first time invested as third order members.
The priest has the option of wearing the cope for the profession and then exchanging it for a maniple and chasuble or he can wear the chasuble and then put on the maniple for Mass.
The same principle applies to those receiving the Sacrament of Matrimony in the Extraorinary Form, with those rites taking place before Mass along with the extra blessings upon the newlywed.