He should have had his phone with him and the cops would have been able to figure out where he was at any time.
She was about to ask him about his phone, but before that, he continued.
"...I remembered turning to leave. But I don't know... I must have passed out..." he placed his hand over Rory's shoulders as she widened her eyes. "I was very drunk..."
Rory wanted to shout at him. If someone else was saying this same story to her, she would roll her eyes and think that was very convenient for him to pass out right at that instance when Laura Brown disappeared.
But this is Liam. He should be telling the truth.
"How long were you out?" Rory asked.
"For more than 20 minutes... 30 minutes, probably..?" Liam said. "I know, Cherry. I know what this sounds like. But~"
"What happened next? Where were you when you woke up?" Rory asked.