Smith hit nine boundaries to follow the century scored against the same opponents in the County Championship match which Hampshire also won yesterday.
Veteran pace bowler Cardigan Connor took three early wickets as Derbyshire slumped to 57 for 5, a position from which they never recovered.
They owed the respectability of their 160 for 9 to their acting captain, Kim Barnett, who finished 52 not out
Connor finished with 3 for 27 and Derbyshire's total never troubled Hampshire despite the early loss of their opening batsman John Stephenson.
Smith featured in three important partnerships, first with Giles White and Paul Whitaker, then with Dimitri Mascarenhas.
The Derbyshire spin bowler Glenn Roberts was rewarded for a tight spell with figures of 2 for 21.