Now that the Twentieth Amendment makes the new session of Congress begin on January 3rd, which is 17 days before the inauguration on January 20th (previously, the new session began, and inauguration occurred, on March 4th), can the new Congress send bills to the outgoing President to sign before January 20th?
Yes, the new Congress can send bills to the President which he/she can sign into law. See Presidential Transitions
The President’s authority to exercise power begins immediately upon being sworn into office and continues until he is no longer the officeholder. By the same token, while congressional oversight of the executive branch is continuous, some activities may take on special significance at the end or beginning of an Administration.
In addition from Wikipedia:
A term of Congress is divided into two "sessions", one for each year; Congress has occasionally also been called into an extra, (or special) session (the Constitution requires Congress to meet at least once each year). A new session commences on January 3 (or another date, if Congress so chooses) each year.
So conceivably the new Congress once in session could pass a bill to be signed by a lame-duck President into law.