Yes, you can only accrue 172 days of service time in a season but there are something like 182 actual days in a season. If you keep a player down for ~11 days or something like that then you basically prevent them from becoming a free agent after 6 years (because their service time will be something like 5.171 and players only become a free agent the offseason after eclipsing 6.000).
Also, players in the top 22% for service time of the 2.XXX service time group at the end of a season get "Super Two" status and go through arbitration four times instead of 3. This past offseason the Super Two cutoff was around 2.133, any player with more than that got to go into arb1. Hutch at 2.128 just fell short of Super Two status.
So, here are the basic models for player control:
Normal - call up on opening day. Get player for 3 years at minimum (plus often an initial September call up to cut their teeth) and 3 years in arbitration.
Prudently milking a year of control - call up player ~11 days after opening day. Get player for something like 2.171 years, technically not 3 seasons, but functionally it is 3 seasons at league minimum + 4 arbitration years (because they will be Super Two).
Dollars over winning - call up player some time in July so they are just under the super 2 cutoff. You get ~3.128 technical years from the player at league minimum + the 3 standard arbitration years.