The fact that so many people still smoke in public places suggests that we may need nationwide compaign to raise awareness of the risks of smoking.

没错，第一个 that 引导的是同位语从句，修饰 the fact。

第二个 that 引导的是宾语从句，用作动词 suggest 的宾语。

句意：这么多人仍然在公共场所吸烟这一事实表明，我们可能需要在全国范围内开展一场运动来提高人们对吸烟危害的认识。