The court jester held what was called a bauble (what you mentioned as a scepter) on occasion. What we think of as jester clothing comes out of early 14th Century Europe and became the "court dress" for jesters well up into the 16th century at least. Here is a jester wearing his hood with bells on the sides from about 1350. He's also wearing his underpants (slops) and hosen.

Another example of a jester, this time from the early 15th Century with other figures around. You can see how different the jester's clothing is. There are a lot of bells and it's much more closely resembling the jester we still think of.

In this late 16th Century painting of a Village Fair there is a jester wearing a particolored (one color one side, a different color on the other) outfit with the traditional hood and bells on it. He's right front and slightly right of center.

However, by the 16th Century, the traditional clothing was falling out of favor. Will Somers, Henry VIII's jester is typically depicted in conservative but fashionable dress for the time.