You have to accept the client order and have it in your client order list in order to send the aux on searches. Just don't turn in the client order if you want to keep on sending it.
Those greyed out client order names on your first image are indicative of you looking at the client orders the NPC has to offer, and showing that you already completed them today. Rather than client orders you currently have accepted.
Example of my currently accepted client orders:
Note how I haven't actually turned in the ones that are marked as done/complete. I just keep them and keep sending my aux on searches.