"most cordial" is a correct and usable phrase in written English.
It is used to describe a warm or fond feeling of welcome. For example, "He greeted us with a most cordial smile."
All involved were most cordial.
Most cordial regards from yours very truly, Klein.
"Even the Harvard players have been most cordial.
(Flint, incidentally, is on most cordial terms with FitzSimons).
accept with most cordial greetings my best wishes for an excellent vacation.
The meeting between hosts and guests was of a most cordial nature.
Cooper is one of the most thoughtful, cordial and well-prepared members of the House.
Most were cordial but curt.
Francis Pretty, one of Francis Drake's men, who tasted this for the first time in 1578, described it as a "clear liquor, which… you shall not only find… very delicate and sweet, but most comfortable and cordial".
Most "defenses" are cordial affairs, not debates, though you should expect to be pressed and argued with regarding your methods, your conclusions, and other aspects of your work.
Political war, after all, is not cordial, and most South Dakotans are.
