They always envisioned that the country would spread from coast to coast, from sea to shining sea. Okay, but how did they know that, if they traveled west far enough, they would eventually come across water and not just infinite land? Because for them to know where the Pacific Ocean begins, that would imply that they've already gone through and settled that land. How did they know about the Western United States and the Pacific Ocean without first claiming all that land?
TL;DR: The timeline of your question is out of order a little. Manifest Destiny was a mid 1800s American idea and there were some maps by then showing the western ocean.
There were halfway decent maps of North America available well before the Manifest Destiny movement so there was no unsolvable mystery at that time as to the extent of the western lands. The Lewis and Clark expedition departed in 1804, at that time those guys went from Missouri all the way to the Pacific Ocean in what is now Washington state. Their expedition was completed round trip to St Louis in 1806. They made lots of maps and others followed in their footsteps to a certain extent. I'm on mobile but I have sources I could get to If needs be. If I remember my college classes properly Manifest Destiny didn't pick up steam as an idea till after 1850.
Hi, it might be worth looking into European explorers who had been mapping the Pacific coastline of North America, others searching for various routes around or through the Americas (e.g. the NorthWest Passage, the Lewis & Clark expedition), the extent of the fur trade (e.g. the Russian-America Company, Hudson's Bay Company, North West Company), and settlements that had already been established on the Pacific. Here are a smattering of earlier posts on some of this to get you started:
Why the lack of exploration of the west coast of North America between the 1500s and 1700s? - featuring /u/Itsalrightwithme. Follow their links to posts on later time periods
Are there any maps of the Spanish controlled Americas that showed their actual areas of influence, rather than territories they simply claimed? featuring /u/RioAbajo
What evidence do we have of pre-1778 contact with Hawaii? featuring /u/ArcturusFlyer
What made cartographers and explorers believe that there was a 'Northwest' passage to be found?
In the early colonial period of the coastal PNW, what other people were there besides Europeans and the coastal natives?, Did the United States know of Russia's presence in the Americas before the Lewis and Clark expedition? , and How aware/concerned were the Western European nations about Russia's operations in Alaska and the Pacific coast of North America during the 18th century? Was there any contact between them? - featuring /u/The_Alaskan
Russian Colonialism in North America - featuring /u/Volush
What changed around 1600 that enabled the French and English to begin colonizing? - featuring /u/jschooltiger