Maintaining long residence times during hydrothermal digestion of cellulosic biomass solids may be complicated by a number of factors, including biomass compaction. Advantages in this regard may be realized by digesting cellulosic biomass solids in an inclined digestion unit. Such methods can comprise: introducing cellulosic biomass solids to a hydrothermal digestion unit comprising one or more inclined surfaces therein; introducing a fluid phase digestion medium containing a slurry catalyst to the hydrothermal digestion unit, the slurry catalyst being capable of activating molecular hydrogen; supplying an upwardly directed flow of molecular hydrogen from a source disposed along each inclined surface as the cellulosic biomass solids descend along each inclined surface; and heating the cellulosic biomass solids as they descend along each inclined surface in the presence of the slurry catalyst and the molecular hydrogen, thereby forming an alcoholic component derived from the cellulosic biomass solids.