How do you fully explore level 1 of the VVVVVV demo?

After spoiling myself and watching a few videos on Youtube I can say that yes, that's all that can be explored in the demo version.

What the demo fails to explain is that actually you start a "level" by reaching its starting sector in free roam through the game's world. That is, the full version does not yank you to a level selection screen after rescuing the first crew member. The opening in the pink area allows for further exploration.

The answer is "yes". The area you cannot access is Space Station 2, as time trial mode calls it. Every yellow area on this map that you cannot access is part of Space Station 2.

It could be confusing however because Space Station 2 shares a few rooms with Space Station 1 (like "B-B-B-Busted", where the area below is simply a harder version of Space Station 1) and also has yellow color. The most noticeable example is "Exhaust Chute", where you simply fall in Space Station 1 and have to use moving platform in Space Station 2.

In fact, you can access Space Station 1 from Space Station 2 by falling up. It's not possible the other way because of spikes (unless you use invincibility mode, but that's not available in the demo).