May I use R710's new Intel 10G Ethernet's second port to connect additional servers into my network?
Your question lacks detail. Both SFP+ ports on the servers represent different NICs. With that chain you need the servers to bridge or route. This may be possible but host configurations are off-topic here.
What is the best way to configure them without buying a new switch?
"The best way" depends on your requirements you've not told us about. The chained setup in your diagram has multiple single points of failure and may be less reliable than you require.
Consider the cost of failure in your setup (a server fails and needs repair or replacement) and reconsider buying a switch.
Best practice for high availability would be to use two 10G switches and connect each server and router to each switch.