To quote Gwen from Historic Enterprises
"Let's not be sucked into that reenactor's myth that "in the 14th C. everybody wore a coif. Coifs start going out of fashion by about 1350 or so and by the 1380's they had become a vestigal part of the ceremonial uniform of the seargentry." According to: "Chaucer and Costume: The Secular Pilgrims in the General Prologue", LF Hodges, D.S. Brewer, Cambridge, 2000, pg.107:
The coif, fashionable for all gentle[men] in the late thirteenth century by 1387 was worn only by some clergy, by helmeted knights (coif de toile), and by serjeants required to wear it in public, in court, and, by special privilege, in the presence of the king. A serjeant's coif denoted his extensive education in the law and his subsequent authority and rank; it was bestowed in a dignified ceremony followed by an elaborate procession and feast. Writing in the fifteenth century, Fortescue names the 'coif of white silk . . . the primary and principle of the sartorial insignia with which serjeants-at-law are decorated at their creation'.
Footnotes to this passage include Newton's "Fashion in the Age of the Black Prince, Newton's "Medieval Costume in England and France, the 13th, 14th and 15th Centuries"
So, hoods are okay for late 14th C., but coifs are right out. For the lower/laboring classes coifs are OK to about the 17th C. Although hoods are almost de rigeur, the late 14th C. and early 15th C. is a period conspicuously devoid of headcoverings for men. Hats will become an absolutely essential component of fashion by 1430 or so, becoming more absurd and ridiculous with each passing year, but we are planning to do a "bare-headed" period."