Actually, it seems like I didn't express myself clearly.

According to my tests, > and -c saves all output of a single gmp-ecm run.

Example:
echo 2^^^^1061-1 | ecm -c 5 1e4 > results.txt

Contents of results.txt
GMP-ECM 6.0.1 [powered by GMP 4.1.4] [ECM]
Input number is 2^1061-1 (320 digits)
Using B1=10000, B2=1186831, polynomial x^1, sigma=2999931604
Step 1 took 783ms
Step 2 took 844ms
Run 2 out of 5:
Using B1=10000, B2=1186831, polynomial x^1, sigma=3364805730
Step 1 took 901ms
Step 2 took 876ms
Run 3 out of 5:
Using B1=10000, B2=1186831, polynomial x^1, sigma=4143695723
Step 1 took 1020ms
Step 2 took 841ms
Run 4 out of 5:
Using B1=10000, B2=1186831, polynomial x^1, sigma=2441763206
Step 1 took 885ms
Step 2 took 879ms
Run 5 out of 5:
Using B1=10000, B2=1186831, polynomial x^1, sigma=523117962
Step 1 took 870ms
Step 2 took 811ms

When you restart gmp-ecm, the contents is lost, of course.