Basically you not only want to sort by org you want to group by org. Meaning display or name and the entries under the org, yes?
Probably the easiest solution is to filter by org.
Use multiple shortcode, each filtering by a different org and above each shortcode add the org name a rough example:
[connections organization="Org 1"]
[connections organization="Org 2"]
You directory seems to be small so this might be a workable solution?