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Its possible to reserve separately or together.
Which is the best course of action depends on how planned you are.
If you know how much time you wish to stay between trains, you can reserve both trains at the same ticketing office and be sure you have your places ready.
If you are unsure how long, then you should reserve the second leg on the spot when you are ready to leave.
Keep in mind that outside the busy holidays (golden week, obon, etc) there is usually no need to reserve seats as the trains are not full. Note that this might not apply to the line you are taking, verify its popularity before taking a decision.