Which of these three sentences is correct?
He snuck a souvenir into his shirt.
He snuck a souvenir under his shirt.
He snuck a souvenir inside his shirt.
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All are possible, but they mean slightly different things.
Some examples, that I think make these distinctions clearer (emphases mine):
With a sly grin, the girl tucked the money into her blouse. —The Patriot Paradox by William Esmont
She stuffed the pillow under her shirt and examined her profile in an attempt to see what she'd look like fat with a baby. —Enchanged Heart by Felicia Mason
Before he walked in the boardinghouse, he tucked the small puppy inside his shirt. —Timberlake by Grey Stone
Note that in the third example (inside) it's unclear whether he tucks the puppy in from the top or the bottom; what's emphasized is where the puppy ends up.
'Inside' and 'into' both imply, to me, that you are sneaking something into the fibers of the shirt fabric. They're not incorrect grammatically, and if you said them someone would know what you meant, but 'under' is probably the best choice here.