@Kaduufrrj I couldn't tell you for sure but I think the phrasing is slightly off. Like they made it sound like there are two baths and they swap at different parts of the day (I really can't imagine what the point of that would be) but I'm pretty sure there's just one bath and there are different designated times for men and women to use it. Which would be most likely if the spring is super old-school, since until the 1860s or so hot springs were unisex and an inn wouldn't have more than one bath. Now days a lot of places still do mixed baths (mostly for couples and/or families) but during those times swimsuits are required.
@Buttzilla A bath with different times for men and women makes sense, but in the mangas usually are two baths with different times for men and women, no logic.