Joseph Smith was trying to translate a language that He did not know. This was the language that the Nephites used to communicate one with another in Ancient Meso-America, long ago, and far away from where Joseph Smith lived.
Guatemala and Southern Mexico is obviously a long way from Upstate New York.
So how was Joseph Smith able to translate the Golden Plates???
Wouldn't it have been great if Joseph Smith had a dictionary so that he could translate the Golden Plates and so that everyone in the World could have available to them THE Book Of Mormon Another Testament Of Jesus Christ???
It turns out that God did provide a dictionary for Joseph Smith to look up words in Ancient Nephite, and then know what the word meant in English.
If you have a portable computer, or a kindle device, or a Black Berry, then you might do what Joseph Smith did. Use it to translate a language that you do not know.
A television or a computer screen shines light from the inside. So to see it easier dimming the lights is advisable. Joseph Smith did much of his translating work in the day time. So to look up a word he used something similar to what the ancient Israelites called a Urim and Thumim which is a miraculous device based on God's technology that is much more advanced then human technology will ever be.
Initially Joseph Smith had to look up the definitions of words, this was however a cumbersome process. Therefor Joseph Smith memorized as much of the language of the Nephites as he could, as he went through the translation process.
More and more as he translated, he did not need to look up definitions of words, a very time consuming process. During the translation of the first one hundred and sixteen pages that were stolen by Martin Harris's wife and her co-conspirators, Joseph Smith out of necessity had to use the translating device provided by God frequently.
Then for a few pages Emma Smith (Joseph's wife) helped out, and was the scribe to Joseph Smith. For whatever reason this was taking too much time. It was then that Oliver Cowdry volunteered to help Joseph Smith, by being the scribe.
After this happened, in two short but very busy months, The Book Of Mormon Another Testament Of Jesus Christ that is now published around the World in numerous languages, was completely translated into English.
There is nothing significant about the hat, except that it might have given some shade, or prevented someone from seeing the translating device that either was a Urim and Thummim or was something similar to a Urim and Thumim (sometimes refereed to as a philosophers stone by the ancients).
God has much better technology than humans did back then or do now, and God is a lot smarter than humans, in fact God is infinitely smart.
So there is no reason that God would not have provided some heavenly technology that is very common in Heaven, because Heaven is more certainly the ultimate information society that there is, as other revelations from God received by Joseph Smith in the Doctrine And Covenants makes clear.
You can believe the Earth is flat, and that Heaven is flat, and that Hell is flat, and that God is someone who only has primitive Dark Ages technology available to Him, but such a belief is highly superstitious, and denies the majesty and power of almighty God, who has more power than any mortal can ever imagine, and is much smarter than any mortal human can readily comprehend.
Moses parted the Red Sea, using the power of God. Don't get hung up on Moses's walking stick, and don't get hung up on Ancient Prophet's when they use a Urim and Thumim, as recorded in the Bible, and don't get hung up on the fact that maybe Joseph Smith a modern Prophet did some of the same things, that is recorded in the Bible that the Ancient Prophets also did.
Focus instead on the truth the whole truth and nothing but the truth. And focus on the words in The Book Of Mormon Another Testament Of Jesus Christ.
Because the words of The Book Of Mormon Another Testament Of Jesus Christ if you allow them, will persuade you that Jesus Christ is real, and that Jesus Christ gave commandments, and if you repent and turn your heart to Jesus Christ, evidenced by you keeping the commandments of Jesus Christ, then you will gain eternal life in the same glorious Kingdom that Jesus Christ lives in, and that for all eternal time.
God bless.