This firm produced perfins from 1914 to 1917.
I am sorry to report that no perfin covers on company letterhead have been located or reported. However, as a consolation prize I will show you what to look for. The example is an envelope used 12 June 1919 from the same approximate time period, or perhaps a little later, that the company was putting perfined stamps on their envelopes.
New York Life Insurance Company cover without perfin, 1919.
Here are some of the stamps which have NLY perfins.