The Brass Monkey has a nice rooftop bar. Bar Pitti has outside seating but only serves beer/wine.
Brass Monkey @ 55 Little West 12th St., New York, NY 212 - 675 - 6686.
Bar Pitti @ 268 6th Ave., New York, NY 212 - 982 - 3300
268 6th Ave, New York, NY 10014
B Bar and Grill is your best bet. It's not W but it's not E either and has a great outdoor garden area.
B Bar and Grill @ 40 East 4th St., New York, NY 212 - 475 - 2220,
Tabla has outdoor tables but it's not W but it's not "Far East" either. Bryant Park Grill is the same. Not W but Not "Far E" but very nice.
Bryant Park Grill @ 25 West 40th St., New York, NY 212 - 840 - 6500.
Tabla @ 11 Madison Ave., New York, NY 212 - 889 - 0667.
You could be more helpful by suggesting a nice place for the OP with outdoor seating than simply stating where E & W is. Where does E & W start in your book. Bowery can't be more than a block E of Broadway. The gentleman is interested in a nice place with outdoor seating. Surely he won't mind walking a few extra blocks for a nice place to have drinks outside.
Do you want specifically outdoor on the ground level or is a rooftop OK? Early int he evening, Print at the top of the Ink48 hotel (11th at 48th) is not too busy and has great decor/view. However, their cocktail menu is not particularly expansive, so it also matters whether you are looking for a creative/artisan drink or just looking more for the right atmosphere.