Patching Array - The Idea

If you can make all sums in $[1, miss-1]$, and you add a number $x \leq miss$, you can now make all sums in $[1, miss + x - 1]$. Why? You already have $[1, miss-1]$.

Adding $x$ gives you $[x, miss-1+x]$. Since $x \leq miss$, these ranges overlap and combine to $[1, miss-1+x]$. The greedy choice: if $miss$ is not covered, add $miss$ itself.

It doubles your range because before you have $[1, miss-1]$, after you get $[1, 2 \cdot miss - 1]$. These ranges are consecutive because $miss$ is the first missing number, so no gap exists.