We analyze measurements of the 0.5-1.0 MeV/nucleon H/He intensity ratio from the Ulysses spacecraft during its first (1992-94) and second (1999-2000) ascent to southern high latitude regions of the heliosphere. These cover a broad range of heliocentric distances (from 5.2 to 2.0 AU) and out-of-ecliptic latitudes (from 18(degree)S to 80(degree)S). During Ulysses' first southern pass, the HI-SCALE instrument measured a series of enhanced particle fluxes associated with the passage of a recurrent corotating interaction region (CIR). Low values ((approximately)6) of the H/He ratio were observed in these recurrent corotating events, with a clear minimum following the passage of the corotating reverse shock. When Ulysses reached high southern latitudes (greater than (degree)S), the H/He ratio always remained below (approximately)10 except during two transient solar events that brought the ratio to high (greater than 20) values. Ulysses' second southern pass was characterized by a higher average value of the H/He ratio. No recurrent pattern was observed in the energetic ion intensity which was dominated by the occurrence of transient events of solar origin. Numerous CIRs, many of which were bounded by forward and reverse shock pairs, were still observed in the solar wind and magnetic field data. The arrival of those CIRs at Ulysses did not always result in a decrease of the H/He ratio; on the contrary, many CIRs showed a higher H/He ratio than some transient events. Within a CIR, however, the H/He ratio usually increased around the forward shock and decreased towards the reverse shock. Throughout the second ascent to southern heliolatitudes, the H/He ratio seldom decreased below (approximately)10 even at high latitudes (greater than 40(degree)S). We interpret these higher values of the H/He ratio in terms of the increasing level of solar activity together with the poor definition and short life that corotating solar wind structures have under solar maximum conditions. The global filling of the heliosphere by transient solar events and the fact that in 1999-2000 Ulysses observed only intermediate (less than 650 km s(sup (minus)1))) solar wind speed (whose contents in pick-up He is less energetic than in the fast solar wind streams observed in 1992-1994) favored the protons with respect to alpha particles. Hence the fact that the average values of the H/He ratio observed by Ulysses during the rising phase of the solar cycle (1999-2000) were higher than those observed during the declining phase (1992-1994).