Where, how and how long will you travel in Hakone?
Is Hakone Freepass highly likely to pay off?
We would like to travel from Tokyo (Shinjuku) to Hakone. Is Shinjuku Station the nearest Odakyu Station?
Our return trip we would like to be from Hakone to Tokyo Station. What will you do after getting to Tokyo Station?
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The only alternative is a return trip from Odawara Station. Beyond Odawara Station is the free ride area for Hakone Freepass travelers.
("Return trip" is ambiguous; here this poster seems to use it for a round trip, while you use it for your way back only.)
You could easily travel between Shinjuku Station and Tokyo Station on your return using an IC card. In that case, it sounds easier to use a transfer gate in Shinjuku Station between Odakyu and JR, where you first put a Hakone Freepass into the machine then touch the IC card to the reader-writer device.
You have some options regarding transportation mode and pass.
(A) You get back to Shinjuku Station with Hakone Freepass, pass through the Odakyu-JR south transfer gate (on the concourse crossing over the tracks), then travel to Tokyo Station by Chuo Line (pronounced like chew-oar-line).
https://www.jreast.co.jp/e/stations/e866.html(B) You choose Hakone Freepass which is to be valid only inside the free ride area beyond Odawara Station, then travel to Tokyo Station either by (non-Shinkansen) Tokaido Line or by Tokaido Shinkansen.
You can purchase a Hakone Freepass
in advance at a counter of Odakyu Sightseeing Service Center, Shinjuku or,
on your first boarding day at a ticket vending machine of the Odakyu station which you travel from and back to.
http://www.odakyu.jp/english/center/#tabitem-02
