Well, if 60+ years is "new", then yes. I believe that it came with the same amendment after Roosevelt died that limits the POTUS to two terms.
That would make it a few years after Adams. If the Johnson you mention is Lyndon, then he never served in Congress after being POTUS. He refused to run for what would, legitimately, have been his second term and died about 4 - 5 years after leaving office. If you mean Andrew Johnson, then same answer as for Adams.
The XXII Amendment applied only to limit a person from holding more than two terms as President, not to any other office. I am unaware of any law ever passed that would bar a President from seeking and holding lesser office.
By the way, Taft served on the Supreme Court after being President. That rounds out the group of Presidents serving in offices after their terms in the White House. Just because it has been a while since anyone has done it, doesn't make it illegal. Although, Bill Clinton is reported to consider Hillary's old Senate seat every now and then.