[[Dear Sister, when a person is going through the discernment process of becoming a diocesan hermit can the cowl be worn?]]
Presuming you mean initial discernment with a diocese prior to admission to any profession, the simple answer is no. The cowl is only given with perpetual profession and then only when the diocesan Bishop grants the cowl canonically. (Not every Bishop does so and some hermits decide to use a different prayer garment.) A simple personal prayer garment can be used in private during the discernment period but cannot be worn publicly since this would imply public standing, rights, and obligations. Similarly, such a garment is NOT given to one by the Church but instead is privately or self-assumed.
Remember that a candidate for possible profession under c 603 is usually a lay person with the rights and obligations of any lay person. Because they are not being incorporated into a community in stages (postulancy, novitiate, juniorate, perpetual profession) with commensurate legal (canonical) rights and obligations they are solitary individuals bound "only" by their baptismal commitments. (I do not disparage such commitments by using "only" here. Baptismal commitments are extremely significant but public profession and consecration imply added canonical rights and obligations.)
Moreover, discernment as to whether one is actually meant to be a hermit of any sort can take a number of years. It simply makes little sense for such a person to be given permission to wear representative eremitical garb when they are neither hermits yet (the transition from lone individual to hermit in an essential sense is something one usually negotiates during discernment and initial formation) nor canonically responsible for the continuation and protection of a vital eremitical tradition. Traditionally, the cowl is associated with the assumption of responsibility for living and representing the fullness of monastic and eremitical life in a formal and canonical sense. When one sees the cowl this is what it indicates or symbolizes. To be faithful to and retain this meaning it is necessary to restrict the ecclesial granting of it to the occasion of perpetual or solemn profession.
I hope this is helpful.