That's not a separate tower, rather it's like a bustle or "tail" attached to the existing tower. I'm making an educated guess here, but right up until the early 60's, building codes required tall buildings to have setbacks. It's how we got buildings like the New Yorker, ESB, or Chrysler. It also explains many mid century buildings too- the Emery Roth and Sons "wedding cake" towers, such as 350 Park Avenue, or 425 Park Avenue. (A Kahn and Jacobs building if we wanna quibble about things)
However, architectural fashion had changed a great deal from the Art Deco era, which lent itself so well to the stepped back towers of the period. Clients wanted sleek, modern towers, and those required setbacks were a problem. So as far as I can tell, the architects found a solution- the law says you had to have setbacks, but it didn't say where you had to put them. So a good many skyscrapers in the late 50's and early 60's were designed so that their towers fronted the avenues, with those legally mandated setbacks positioned to the sides and rear of the building in various creative configurations. This effect served to give the buildings a "tail" of sorts.
Some buildings which did this:
1285 6th Avenue aka UBS Building
399 Park Avenue
1290 6th Avenue (This building might have the biggest tail of the lot, with the most elaborate configuration of setbacks)
1301 6th Avenue
270 Park Avenue
277 Park Avenue
In 270's case, the tail is only really attached by a narrow section, so it's easy to ignore, or mistake for a second building. Others like 1290 and 277 had elaborate tails that wrapped around the main towers and were more visible from the street.
Here's a picture of 277, showing the tail:
(picture from replicabuildings.com)
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