Needing victory to square the series after a comprehensive nine-wicket loss inside three days in the first Test in Dominica, the Caribbean side did not help their cause after early strikes by Jerome Taylor with the new ball.
They wasted at least two clear-cut opportunities to limit the tourists' progress and give their depleted side a better chance of upsetting Test cricket's second-ranked team.
Smith, who will resume on the second morning on 135 after 357 minutes at the crease during which he struck 16 fours and two sixes off 278 deliveries, was missed by Darren Bravo at first slip off the part-time bowling of Kraigg Brathwaite an hour into the day's final session.
Taylor, who finished with three for 18 off 15 overs, was a model of accuracy and efficiency from the start of the day, removing David Warner for a duck in the first over, caught by Shai Hope at third slip, and fellow opener Shaun Marsh leg-before for 11 shortly after to be 16 for two.
It should have been 22 for three when Michael Clarke, on three, popped up a catch back to Kemar Roach off his own bowling.