To find out how many more customer calls Mrs. Gutierrez made in the second year than in the first, we need to subtract the number of calls she made in the first year from the number of calls she made in the second year.
The number of calls made in the first year is 5,842, and in the second year, it is 10,362.
Now, we perform the subtraction:
\[ 10,362 - 5,842 = 4,520 \]
Thus, Mrs. Gutierrez made 4,520 more customer calls in the second year than in the first year.