If your grandfather was living in New York at the time of his induction, try submitting a war records search request form to the NY State Archives. You can download a copy of the form at http://www.archives.nysed.gov/a/research/res_topics_mi_warsvc_form.shtmlhttp://www.archives.nysed.gov/a/research/res_topics_mi_warsvc_form.shtml. You'll also need to send a check for $1.00.
After the war all NY WW1 soldiers filled out service summary cards, and that's what the archives will check. If they find your grandfather's, they'll send a copy--a goldmine of info that includes dates of induction and discharge, units served with (and transfer dates if any), and dates of overseas service from day of departure to day of return. Sounds like your grandfather might have been a NJ soldier, in which case he wouldn't be on file, but for a buck it's worth a try. Good luck!