To send new request to the member you already have conversation with, open their listing and on the right hand side click on the "new request".
You can open the listing or profile from the existing conversation by clicking on the "profile" or "listing" icon at the top of the message:
Once on the listing page, on the right hand side, you will see "continue conversation" button. Just below that you will see "New Request". Click on it to start new conversation.
A new box will open where you can type in your message. You will also be notified that you already have a conversation with this member.
You can find them in messages as 2 different conversations: