Is there a way to let the ARS know that all digits have been entered and to immediately dial out." - Yes, press #, so you've found it.
It all depends on what country you are in (ie. the dial plan for different types of numbers like local, national, international) and the type of external lines. Also what you want to allow different users to dial. You didn't tell us anything about any of that.
Its the ; at the end of short code that tells the system to wait. For things like SIP trunks that's a must as they need to send the full set of digits in one go. The delay is set by the Dial Delay Time and outside the US its just 1 second rather than 4 (never known why US users are thought to dither in dialing more than everyone else). Reduce it if you want but be prepared for kickback from some users.
You also didn't mention what phone type are being used. Many of the Avaya phones can now be set to en-bloc dialing. So the user dials the whole number of the phone as fast or slow as they like, and it only goes to the system when they press Dial or lift the handset. Removes any issues around having a short Dial Delay Time.
Stuck in a never ending cycle of file copying.