Using a stealth weapon always counts as a stealth attack. However:
"Some Weapons have the Stealth trait. ...
The first time each Soldier performs a Stealth Attack during a game turn, he automatically gets a Defeat Cover roll of a 6."
page 12 of the 2nd edition rules
I'd say if you want to use second action as a stealth attack, you have to use a non-stealth weapon first.