Min if not zero with another condition

Hi All,

I am looking for a way to find a min non-zero value in a range with another condition. for example I have a range with names (ColE) and a range with numbers (ColF) so I would like to find a Min value after 0 in ColF for a name in Cell B1 within ColE. I use the following formula to get the Min value after 0 in the whole range. How could I update it to add nother condition to get the non-zero Min value within the name of cell B1. Thanks

Formula:

=SMALL(F1:F10,COUNTIF(F1:F10,0)+1)