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 3 of the 8 sums) and then the next smallest numbers (5, 7, 11, 13) in the positions involved in 2 of the 8 sums, so that $S_1+\dots+S_8$ is minimised. I believe this may be the best possible.

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.

Here's an example that beats the OP's:

  |13  5 11| 29
  |31  3 19| 53
  |29 23  7| 59
  ----------
43 73 31 37  23

I got this by putting the smallest number (3) in the most central position and proceeding from there, leaving the positions only featured in two of the sums (i.e. the squares orthogonally adjacent to the central square) until the end.

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 3 of the 8 sums) and then the next smallest numbers (5, 7, 11, 13) in the positions involved in 2 of the 8 sums, so that $S_1+\dots+S_8$ is minimised. I believe this may be the best possible.