Here's an example that beats the OP's:
|11 41 7| 59 |47 3 17| 67 |13 83 5| 101 ------------ 23 71 127 29 19
I got this by putting the smallest possible number (3) in the most central position (which is involved in four of the eight sums) and then the next smallest numbers (5, 7, 11, 13) in the positions involved in three of the eight sums, so that $S_1+\dots+S_8$ is minimised.