How about creatures (perhaps intelligent) born on the central band which have now adjusted to both extremes. They spend time basking in the sun of the hot side, and then go on hunting trips in the cold side (perhaps because they find the hot side creatures inedible for some reason).
Well, it isn't like there's this terminator line, and on one side of it it is bright hot sun, and just a few feet away it is dark and cold. In a planet with an atmosphere, there's a broad band - full-sun shades to twilight shades to dusk shades to full-night, over the course not of feet, but of tens or hundreds of miles. The temperature will have a similar gradient.
So, if there's an animal that does what you suggest, it probably isn't doing it as a short hop, but as a major journey - go to feed on the night side, come back to breed on the day side, or suchlike.
Which brings me to another note - unless there's a major moon or other body that drives a periodic change, this planet does not have "seasons", as we understand them. The climate would have weather, but otherwise be "steady state" over the course of the year.