Creating dedicated sockets for each transfer is one solution, and it's not a bad one unless the number of simultaneous connections is large (only so many IP ports are available on a system, and the server will need twice as many). Threads don't simplify this as much as you might think, and introduce their own challenges; select is a simpler way to efficiently transfer data on multiple sockets from a single thread/process. It works by exposing the underlying operating system's knowledge of which sockets are ready for reading and writing to the program.
The challenge for you with the multi-socket approach, regardless of threading choices, is that the server will have to tell the recipient to open a new connection back to the server for each new transfer. Now you need a command mechansim to tell the recipient to open a new connection for the next file.
Another option would be to open only one socket, but send multiple files simultaneously over the socket. You might accomplish this by sending a data structure containing the next parts of each file instead of simply streaming the file directly. For example, you might send a message that looks something like this (rendered in JSON for clarity, but it would be a valid transport format):
[
{
"name": "file.txt",
"bytes": "some smallish chunk of content",
"eof": false
},
{
"name": "another.txt",
"bytes": "chunk of another.txt content",
"eof": true
}
]
This example is of course naively simplistic, but hopefully it's enough to get the idea across: By structuring the messages you're sending, you can describe to which files, which chunks of bytes belong, and then send multiple chunks of multiple files at once. Because of your client->server->client approach, this seems like the best path forward to me.