You want to open multiple TCP connections.
You will either need to use an evented loop to handle the back-and-forth of keeping all the pipes full, or need to use threads/processes. As someone commented above, you can use 'fork' to make multiple copies of your script, each one can make one TCP connection and easily keep the connection full. That's probably the simple solution.
If you want to keep your program as a single process, it's a bit more work, but still possible.
If you are opening a LOT of connections, you'll want to read this: http://www.kegel.com/c10k.html
You might also consider using a faster language like C or Go, since using perl does involve some overhead. (I'd do a test first, maybe the overhead is negligible. Test by using a tool like curl
to send a big file to see if it gets higher bandwidth than your perl program.)