For speed and urgency:
* Hurried: Implies haste and a lack of time.
* Hasten: Verb meaning to speed up.
* Scurry: Used for quick, nervous movement.
* Dash: A sudden burst of speed.
* Bolt: To move very quickly.
* Hustle: To move or work with energy and speed.
* Expedite: To speed up a process.
* Accelerate: To increase speed.
For overwhelming demand:
* Surge: A sudden increase in something.
* Flood: An overwhelming amount of something.
* Wave: A large amount of something that comes suddenly.
* Boom: A sudden increase in activity or popularity.
For sudden emotion:
* Elation: A feeling of great joy and excitement.
* Excitement: A feeling of eagerness and anticipation.
* Thrill: A feeling of intense excitement and pleasure.
* Frenzy: A state of great excitement and activity.
For a hurried visit:
* Pop in: To make a quick visit.
* Drop by: To make a brief visit.
* Swing by: To make a short visit.
Other:
* Rush hour: The time of day when traffic is heaviest.
* Rush job: A task that needs to be completed quickly.
To choose the best word, consider the specific situation and the nuance you want to convey.