Yes. 
IIRC Grundy or Golomb might have explored something similar, though I don’t recall the use of subtractions (change giving).
I’ll try to extend it to four coins.
Once you start with a $3 coin, the $14 and $27 appear optimal as a three-coin set. Add a $124 to get the best four-coin set. The coins {3, 14, 27, 124} can produce any sum up to $28, $152, $276, $400 using 4, 5, 6, or 7 coins respectively. Given the {3, 14, 27}, $124 is the coin which optimizes each of the 5-, 6-, 7-coin cases.
But that’s assuming we have a $3 coin. The coins {2, 14, 25, 136} can produce any sum up to $32, $168, $304, $440 using 4, 5, 6, or 7 coins respectively.
That four-coin set might be the best. Other good sets include { 4, 11, 28, 133} which can produce any sum up to $26, $159, $292, $425 using 4, 5, 6, or 7 coins respectively.