After you depart from the island you will have to answer the phone again.
Fast and full achievements guide below:
Timestamps:
00:00 Intro + Case 1: Picked it up Achievement
00:46 Case 1: Tried to leave Achievement
01:02 Case 1: Hustled Hard Achievement
03:08 Case 1: Got Spooked Achievement
03:30 Case 1: Picked a Winner Achievement
03:44 Case 1: Answered the call Achievement
03:56 Case 2: Sticky Achievement
04:32 Case 2: Gross Achievement, Big Spender, Fibber, Expert Investigator, and Snitch Achievements
09:55 Case 3: Made it to town Achievement
10:50 Case 3: Graveyard break in Achievement
11:19 Case 3: Haunted photo, Bad room Achievements
16:40 Case 3: Monkey, and Kiss Achievements
17:34 Case 3: Finishing the game