Science journalist here who has written a lot on de-extinction projects and on Pleistocene megafauna in general.

In the early 1800's this was a perfectly reasonable thing to contemplate. Mammoth bones had been found in both American and Eurasia -- and not all stuff that had turned into stone, like dinosaur bones. Actual bones full of marrow.

The idea of mammoths and mastodons being a type of elephant that is a distinct species from other contemporary Asian or African elephants was only put forward by Georges Cuvier in 1796, a mere eight years before Lewis and Clark set forth on their expedition in which Jefferson had asked them to keep an eye out for evidence of mammoths. The concept of the various species was new, very little was known about them, and America was a large continent that still had a lot of territory and lifeforms unknown to western naturalists.

In 1799 an ivory collector named Ossip Shumakhov first noticed the shapeless mass of what turned out to be the thawing remains of a frozen mammoth in a swamp near the Lena River in Siberia. This was confirmed by a botanist named Mikhail Adams, who later collected hair, skin, flesh and the entire right ear from the carcass.

Put yourself in the shoes of a scientific mind of the day. If you are hearing reports of not only bones but actual soft tissue from what appears to be a recently dead animal, wouldn't you have assumed that the species is still around?

At the time, people didn't know that the Earth had gone through a series of ice ages and intermittent warming periods over the course of millions of years (not that any frozen mammoths are that old) and couldn't be expected to understand that a mammoth might have died right as the Earth grew colder, would stay frozen for tens of thousands of years, and then thaw out due to a warming climate.

Extinction was also still a new and controversial subject. Taxonomy had only just been invented, paleontology basically didn't exist, and words like "dinosaur," "Pleistocene" and "evolution" (referring to the biological process) had not been coined yet. The language and scientific systems to even begin to rationally talk about extinction were only just emerging around 1800 when Jefferson was contemplating the mammoth.

Source: I recommend "Mammoths," By Adrian Lister and Paul Bahn, 1994. ISBN 0-02-572985-3.

Essentially, he didn't believe in extinction.

Jefferson believed in a perfect God. There are multiple points in the Bible that point to a perfect God (Psalm 18:30, 2 Samuel 22:31, Deuteronomy 32:4). In his book Big Bone Lick, Stanley Hedeen says that extinction conflicts with "the prevailing Judeo-Christian belief in a perfect, unchanging creation" (38). To Jefferson, a perfect God would not create a species that would go extinct.

Therefore, to Jefferson, the bones being unearthed in KY and NY must belong to a living species. He assumed that they must be living in the unexplored regions of the US.

Jefferson believed that species eventually identified as mastodon were a species related to an elephant. In his 1797 presentation to the American Philosophical Society, Jefferson explained that the elephant-like creatures relocated to the States because of climate change. He said, "... consequently that on our earth there has been a time when the temperature of the poles suited the constitution of the elephant, the rhinoceros, and the hippopotamus: and in proportion as the remoter zones become successively too cold, these animals have retired more and more towards the Equatorial regions, till now that they are reduced to the torrid zones as the ultimate stage of their existence" (257).

In this same presentation, Jefferson provides three reasons as to why the Megalonyx (giant sloth) was not extinct. First, he states that the large predatorial animals of Africa live in the uninhabited part of the country, so it's possible that the Megalonyx would live in the uninhabited part of the United States. Second, he quotes first-hand sightings of the creature from the sixteenth century until his present day. Third, he claims that it is "probable that the great-claw has at all times been the rarest of animals" (256).

He eventually changed his view of the concept of the perfect God and accepted extinction as a reality. In an 1823 letter to John Adams he states "It is impossible, I say, for the human mind not to believe that there is, in all this, design and cause effect, up to an ultimate cause, a fabricator of all things from matter to motion, their preserver and regulator while permitted to exist in their present forms, and their regenerator into new and other forms" (qtd in Hedeen 103).

Sources: Hedeen, Stanley. Big Bone Lick. 2011. Jefferson, Thomas. "A Memoir on the Discovery of Certain Bones of the Quadruped of the Clawed Kind in the Western Parts of Virginia." Transactions of the American Philosophical Society 1797. Vol 4. Pgs: 264-60

(I wrote my thesis on this and am thrilled a question finally popped up in my tiny area!)