If you're buying travel insurance, you would want to cover your entire trip, not for just medical expenses, but for the other exigencies that such insurance can cover: loss, theft, travel or baggage delays, emergency cash, legal assistance.
If you buy medical insurance only, you may not have to cover your entire trip. However, not doing so could present problems if, during the 'non-covered' period, something happens. As importantly, should you buy adventure sports insurance, and only for the period in which you'll engage in such activities, a subsequent claim could be rejected if your coverage period is less than the period of your trip. Look carefully at the provider's terms and conditions to ensure that you're protected fully.
Another option is to add coverage while on your trip. World Nomads, for example, does just that. As it also has the usual travel insurance, you may be able to combine both, with an activity upgrade for a portion of the trip. (I'm sure there must be other similar companies; I'm not affiliated with it in any way; this was the result of a Google search.)
Check whether you have protection currently where you live, e.g., through your employer, which would cover you for medical expenses while abroad. Often these are reimbursements: you pay in full and submit a claim upon your return. And, in the US, a trip to the emergency room could easily cost $3000 (and the ambulance that takes you there... $750 !).
Whichever approach works for you, healthcare in the United States is quite expensive. You're never denied emergency care based on ability to pay: after assessment and, as appropriate, treatment, you are expected to pay for it, whether through insurance or from your pocket.
