You can use -b to specify a cookie file to read the cookies from as well.
In many situations using -c and -b to the same file is what you want:
curl -b cookies.txt -c cookies.txt http://example.com
Further
Using only -c will make curl start with no cookies but still parse and understand cookies and if redirects or multiple URLs are used, it will then use the received cookies within the single invoke before it writes them all to the output file in the end.
The -b option feeds a set of initial cookies into curl so that it knows about them at start, and it activates curl’s cookie parser so that it’ll parse and use incoming cookies as well.
See Also
The cookies chapter in the Everything curl book.