As macraf notes in his answer, all Shinkansen services north of Morioka (Hayabusa and Hayate) have only reserved seats. Between Morioka and Tokyo, you can take a slower Yamabiko train, which will have unreserved seating.
If you want to exit the ticket gates at the intermediate station
This is the JR definition of a “stopover” (途中下車). Basically, base fare tickets allow stopovers (unless it’s for a very short trip inside an urban area) but limited express tickets (either for reserved or unreserved seats) do not.
So in this case the best strategy is to buy a single base fare ticket from Hakodate to Tokyo, and two separate limited express tickets. When you exit at your intermediate station, your base fare ticket will be returned, and you can use it again to re-enter. Also, since this is a very long trip, your base fare ticket will actually be valid for several days (the validity period will be printed on the ticket), so you could even make several overnight stopovers.
If you do not exit the ticket gates at the intermediate station
If you are using reserved seats, the above also applies because reserved seat tickets are only issued for a particular train on a particular day, and are not valid on any other train.
If you are using unreserved seats, you can get off a train and on another one with a single limited express ticket because unreserved seat tickets are valid on all trains on a particular day. (For example if I buy an unreserved seat ticket from Morioka to Tokyo valid today, I can stop in Sendai for a bowl of udon, and again in Koriyama to buy the excelent senbei they sell there.)
So in your case, for maximum stopover flexibility you can buy a reserved seat ticket from Hakodate to Morioka (since trains on this section have no unreserved seats), and then an unreserved seat ticket from Morioka to Tokyo (and you can make as many stopovers as you want on this section as long as you don’t exit the ticket gates).
If you buy separate tickets for the journey:
With reserved seats it’s quite obvious: you have to buy two tickets for two legs of the journey. With Shinkansen express tickets it will be more expensive than buying a single ticket from Tokyo to Hakodate.
Example with Tohoku Shinkansen:
Tokyo – Shin Aomori: 17,150 yen
Tokyo – Sendai – Shin Aomori: 11,000 + 11,010 yen
With non-reserved seats you can leave the train and continue your journey on a different train, but you cannot leave the Shinkansen area of the station. If you do, it will either invalidate the express ticket (if you change to regular JR trains called “zairaisen“) or will invalidate the whole ticket (basic fare + express ticket) if you leave the station (ie. ticket area).
If your basic fare ticket was issued for a distance over 100 km you can ask for a refund for the unused part of the basic fare (but you would have to ask for it in a manned gate before leaving the ticket area).
Mind that the fastest trains on Tohoku Shinkansen and probably on Hokkaido Shinkansen provide all-reserved seats.
There’s basically little value in leaving the train, but staying inside the Shinkansen area of the station. You can either buy a box of food (bento) or cheap soba noodles at the platform.
If you travel with JR Pass, then it’s up to you how you split your journey.
With non-reserved seats you can leave whenever you want, with reserved seats you would rather have to ask for the appropriate reservations upfront. Although technically you could get a Tokyo-Hakodate reservation, leave and get another reservation for example from Sendai-Hakodate, I guess JR is trying to prevent this (as well as overbooking and no-shows) by stamping or making notes on the JR Pass.
Credit:stackoverflow.com‘
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