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.

$$457 + x > 700$$

Subtract 457 from both sides of the inequality.

$$x > 700 - 457$$ $$x > 243$$

This means the number added must be greater than 243.

The number she added ($x$) was less than 300.

$$x < 300$$

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 $\text{244}$.