By the end of the 1st century, the apostolic deposit of the diaconate (i.e., special share in the Apostles' ministry, given by the laying on of hands, including sacraments/catechesis/evangelization but not including the "breaking of bread;" with moral and spiritual qualities expected over and above initial conversion/baptism) was taking on the form of a more stable office. Fr. Enright continues the section, "New Testament and the 2nd Century" (p. 9-11 in my copy) by looking at the Didache (prob. Syria, ca. 90 give or take a decade or so), the Shepherd of Hermas (Rome, before 150), St. Ignatius of Antioch (martyred ca. 115), Justin Martyr (Rome, martyred ca. 155), and Tertullian (Carthage, ca. 180-210).
All five of these sources share the concern of 1 Timothy, that bishops and deacons (and priests) have particular moral and spiritual qualities. All of them list bishops, priests, and deacons as ministers of particular importance (sometimes adding others to the list, sometimes just those three alone). All of them repeat the basic parameters of the deacon's ministry, in varying levels of detail. Justin in particular, in his First Apology, esp. Ch. 65-6, gives us one of the earliest descriptions of the Mass, including the deacon's role of distributing Eucharist.
By the beginning of the 3rd century, Hippolytus, a priest in Rome, records for us details of actual ordination rites (p. 13, section "From the 3rd century to the 5th century"). These too reflect the apostolic deposit of Acts 6 etc., and the development of precision since. Bishops, priests, and deacons, and the minor orders, are described. Ritual actions and language to distinguish clearly the grades from each other have developed. There is still plenty of variation in what deacons actually do after ordination, especially in emergencies (read: pagan persecution).
After noting further examples in Cyprian, Clement of Alexandria, Origen, Egeria and Jerome, he starts talking about the decline of the diaconate (p. 16 in my copy; the paragraph that begins "The councils of the ancient church are a vital source...") He notes how conciliar evidence points to conflict between deacons and priests, and how councils started restricting the role of deacons in liturgy, etc. Fr. Enright argues (p. 17, 19) that the expansion of the Church in the 4th century increased "demand" for priests, who could say Mass outside the bishop's own church, and thus simultaneously decreased "demand" for deacons. Deacons, he argues, continued to serve their bishops, especially as legates and administrators; they didn't penetrate the rural communities like the priests did, and therefore faded in relative importance. He doesn't talk about formation and the bishop's trust in a proven deacon, but he implies some of this when stating that the diaconate thus became, by ca. 600, a "transitional" stage to the priesthood.
To me, what's interesting about all this is how stable the diaconate proved to be over these first 5 or 6 centuries. Despite all the changes the Church went through, despite all the different ways deacons were used in practice, despite the changing relationship between deacons and priests, the apostolic core (sharing the bishop's ministry, laying on of hands, sacraments/catechesis/evangelization, not confecting but giving Eucharist, expected moral/spiritual qualities) and most of the 2nd-century development (ordination rite, characteristic liturgical and ministerial roles) are visible at every point along the way. Given that stability, should we really see the acceptance of the "transitional" diaconate as a decline?