Your new hat becomes You. I would combine the Kilt with T-Shirt in a matching colour (light blue or white) and put it into the Kilt.

On the other hand I may have to get familiarize with the sight of a T-Shirt worn over the Kilt. If You realiy like to wear Your T-Shirt this way ... why not.

You ask for my thoughts. I am inexpertly with english but hope I wasn't brutal.