I know. But it can look funny. I just tried my Ihram on this morning for the first time (I am performing Hajj this year) and I looked something like what you described in your post.

The Ihram must be worn by pilgrims performing Hajj. It consists of two unsewn sheets of cotton, one wrapped around the lower body and one draped over the upper body. During the Hajj, a man may undrape his right shoulder but otherwise will be well covered. A pilgrim is required to wear an Ihram because it removes the class, wealth and status and individuality aspects of the pilgrim - making him equal and at one before the masses and more importantly before God.

Generally, one dons the Ihram at their point of departure for Makkah. If I were flying from DFW to AMM and on to JED, I would also change in the lounge at AMM.