It's simpler than you might think.
Look at the map (with tolls removed), decide which roads you do not want to travel on and use an alternative route.
A quick check of the geography shows that alternative bridges are a ways upriver, the I84 doesn't add that much time.
You have to cross the Hudson River somewhere, so its bridges are natural choke points. The three closest to, but not entering, New York City are:
New Tappan Zee Bridge on I-87/I-287/NYS Thruway, toll $5.00 eastbound
Bear Mountain Bridge on US-6/US-202, toll $1.50 eastbound
Newburgh-Beacon Bridge on I-84, toll $1.50 eastbound.
The bridges further north ...