What an interesting question!
In this example, to tuck something is to put it into a (small) space. You can substitute the word "put".
• She tucked a coin into the boy's palm.
• I tucked a napkin under his sandwich.
It's not necessarily a metaphor because she really did put something under her pillow and either she WAS ashamed or she should have been! What was it? A note from her lover? A photo?
Everything up to that line tells us that the person singing the song has reason to doubt her faithfulness, so she was probably up to something!!!