Rain does very much affect scent, and scent distribution. A good question. Key to remember: Scent clings to wet things. Wet plants, wet logs, wet grass.
Simply put, it increases concentration, but does not necessarily limit area. It does stay closer to the ground, yes. In many ways it is easier for dogs to detect odor when the ground is wet, unless they have not practiced in it. That’s why we suggest you do- it will change how odor moves in the environment.
The analogy we often use in seminars is a ‘wet dog’, but it gets a bit science-y. Why do ‘wet dogs’ smell more than dry dogs? On a dry dog all those VOCs are bound up next to the hair follicle and can’t move very easily. They are trapped. On a wet dog the VOCs, which are carried by water (as that is how they transfer) can transfer down the length of the hair and out into the environment with far more ease. They also bind with your scent receptors better.
Also, in part, this is why dog’s noses are wet. Scent particles bind more readily and transfer to other wet objects (this is over simplification, but you get the general idea).