"Hütchen" isn't a hat, it's a friendly sprite/goblin/elf who is named after the distinctive hat he wears - the previous "Chapter" of the collection shows him and his hat Hütchen, and says:

"Ich bin ein Geist, und geh' herum
      und heiße mit Namen Hütchen:
Wer früh aufsteht und fleißig ist,
      bekommt von mir ein Gütchen!"

[I'm a spirit named Hütchen who goes round and helps out anyone who gets up early and works hard]. In the poem he gives the priest a ring made from a bit of his laurel wreath, and this enables the priest to speak Latin at the convocation and beat everybody in rhetoric and argument. The last verse says:

"If only Hütchen would travel round the world and give every stupid person a ring like this, (that would make them clever) ... please come back dear Hütchen and bring a thousand such rings with you!"