An economist....
What free trade enables is for monopolistic corporations to take control of foreign resources to circumvent and undermine US labor laws. They'll buy the rights to the same crops growing in Mexico or Candida for cheaper than the US to undercut US farmers. None of that changes the price they are selling goods in the US and in fact ultimately results in prices going up in both Mexico and Candida.
If it were true that free trade results in more competition, then you'd expect that there would be more companies competing, not fewer. We see fewer companies.
Your theory has been tested in reality and it fails.
More importantly, the raw number of corporations isn't how you measure a monopoly. Profit margins are the key sign of monopoly pricing- but in practice, corporate profit margins have been relatively stable in percentage terms over time. Prices for virtually all consumer goods are way down in real terms over the last 30 years.
In particular, trade might get you more competition with fewer corporations, but via competition with overseas producers. These might show up as a few importing corporations, representing a whole slew of competitors abroad.