You need to check with your NY club to see whether your club has reciprocal arrangements in other cities.
If you are visiting somewhere else for a short period of time, you can always inquire whether you can purchase a mini-membership for a week or so and then do the math whether it is financially advantageous. Clubs will often allow out-of-towners to be guests more often even if their standard policy says that you can only do so once per quarter. This usually applies to in-town guests.
If your club or another club you will be visting is a participating member of IHRSA Passport, you may be entitled to a discounted day/week rate.
Here are some links for IHRSA:
Copied from IHRSA:
The International Health, Racquet & Sportsclub Association (IHRSA) represents health and fitness facilities, gyms, and sports clubs worldwide. IHRSA and its members are dedicated to making the world healthier through regular exercise and fitness promotion. If you own, operate or work in a fitness club and would like to learn more about how IHRSA and our global activities can benefit you and your fitness business, please visit www.ihrsa.org.
Healthclubs.com, IHRSA’s consumer web site, makes it easier than ever for anyone to locate IHRSA member health clubs in the United States, Canada, and internationally. Fitness minded men and women can visit the site to find quality health clubs at home and while travelling. Healthclubs.com displays maps with the location of clubs in given communities. Visitors can also search for clubs that offer specific programs or amenities (e.g., tennis courts, swimming pools, etc.). Individuals who are traveling can access www.healthclubs.com to find clubs that participate in the IHRSA Passport Program and find information on guest fees.
Hope this Helps!