Step 1: Coverage distance of a tower.
The line-of-sight range of a TV tower of height $h$ is \[ d = \sqrt{2Rh}, \] where $R$ is the Earth's radius. So the range grows with the square root of the height.
Step 2: Set up the doubling condition.
The first range is $d_1 = \sqrt{2Rh_1}$ with $h_1 = 100\,\text{m}$. We want the new range to be double: $d_2 = 2d_1$.
Step 3: Relate the heights.
Writing $d_2 = \sqrt{2Rh_2}$ and setting it to $2d_1$, \[ \sqrt{2Rh_2} = 2\sqrt{2Rh_1}. \] Squaring both sides gives $h_2 = 4h_1$.
Step 4: Find the new height.
\[ h_2 = 4 \times 100 = 400\ \text{m}. \]
Step 5: Find the increase, not the new height.
The question asks how much taller it must be, so we subtract: \[ \Delta h = h_2 - h_1 = 400 - 100 = 300\ \text{m}. \]
Step 6: Conclusion.
The tower height must be increased by $300\,\text{m}$. \[ \boxed{300\ \text{m}} \]