Not a transit map in the usual sense of carrying passengers, this map instead depicts a network that conveyed mail at speeds of up to 35mph under the streets of New York from 1897 to 1953 (barring a small gap during World War I when it was shut down to conserve funds for the war effort).
This map probably shows the system at its height pre-WWI, with over 27 miles of tube. Even then, the costs of running such a system were becoming prohibitive, and the new-fangled automobile was becoming a viable and cost-effective alternative to transporting mail across the city.
New York wasn’t alone in having such a system, although it carried more mail than most: a single canister could hold up to 600 letters. Paris’ pneumatic tube mail (see a 1967 map here) remained in service until 1983, when it was finally ousted by fax and telex machines (remember those?)
Source: via Untapped Cities, diagram originally from “The Works” book by Kate Ascher)
I have the book that map is sourced from. An excellent book on urban infrastructure in general, and New York City in particular!