https://github.com/Workfront/workfront-api-constants/blob/master/src/constants.ts keyed me into a bunch of stuff, like the OR:x:condition syntax and the _Mod expression modifiers and things like "between" for date ranges. The Adobe documentation then added more color to it.
I'm not fully following your desired boolean expression, though. It may be helpful to write out the boolean as something like this first:
(DATE >= 2023-01-01 AND DATE <= 2023-07-01) OR (DATE >= 2024-01-01)
From there, you can figure out how to do this in the API:
hour/search?entryDate_Mod=between?entryDate=2023-01-01&entryDate_Range=2023-07-01&OR:1:entryDate_Mod=gte&OR:1:entryDate=2024-01-01
Note that the API syntax does not support all types of boolean conditions that you can write out.