Tilt the card $45^\circ$ and center it on a corner where four grid squares meet, as shown. Because the card's diagonal, $\sqrt{1.5^2 + 1.5^2} = \sqrt{4.5} \approx 2.1,$ is longer than $2,$ each of the four corners of the card reaches past a grid line into the next square.

The card covers the central $2 \times 2$ block of $4$ squares and pokes into $2$ more squares on each of its four sides, giving $4 + 4 \times 2 = 12$ squares.

This is also the most possible. The card is only $1.5$ inches wide, so its overall width and height are each at most $2.1$ inches; it therefore lies within a $4 \times 4$ block of $16$ squares. Its four pointed corners are the only parts that reach the edge of that block, so it can never reach the four corner squares of the block, leaving at most $12.$ Since $12$ is achievable, the maximum is $12,$ which falls in the range "$12$ or more."

Thus, the correct answer is E .