Set up the inequality based on the problem statement.

Step 1: Set up the inequality based on the problem statement. Joy starts with 457 and adds a number (let's call it $x$) to get an answer more than 700. $457 + x > 700$

Step 2: Solve the inequality for $x$. Subtract 457 from both sides of the inequality. $x > 700 - 457$ $x > 243$ This means the number added must be greater than 243.

Step 3: Consider the second condition given in the problem. The number she added ($x$) was less than 300. $x < 300$

Step 4: Combine both conditions and find the smallest integer. We have two conditions for $x$: $x > 243$ and $x < 300$. This means $x$ must be an integer between 243 and 300 (exclusive). The integers satisfying $x > 243$ are $244, 245, 246, \dots$. The smallest integer that is greater than 243 is 244. This number, 244, also satisfies the condition $244 < 300$.

The smallest number it could be is $\boxed{244}$.