Whether characters wear hazmat suits is situational, not dependent on which season they went to the Upside Down.
Throughout the series, people who have access to hazmat suits (and the time to put them on) do wear them. US government scientists/soldiers (and Hopper and Joyce) in Season 1, and Russians in season 3 all wear hazmat suits in and around the Upside Down. During each season, some characters enter the Upside Down without hazmat suits, either because they went unexpectedly or because they did not have time or access (or both) to put on a hazmat suit when driven to the Upside Down by necessity.
When Will was saved, the main characters learned that long term exposure to the Upside Down wasn't great for your health (though also not lethal or unrecoverable). The government already knew this when it started sending people into the Upside Down and thus used hazmat suits because it had the resources to do so. This would enable the same people to go into the Upside Down over and over without the long term effects. Nancy and Jonathan went to the Upside Down in Season 1 without protection because they were exploring and did not know any better, and because they did not have access to hazmat suits.
The US government operation to explore the Upside Down ends after Season 1. At that point, in season 2, the only characters going into the Upside Down are either going unintentionally or intentionally for a very specific purpose that they believe (or hope) will take a very short time. Nobody goes there thinking they'll be gone for weeks like Will was. They are also generally do not have access to hazmats suits or the time to go get them. Driven by necessity, they accept the risks of exposure (if they know about them) and count on keeping that exposure as brief as they can.
In Season 3, the Russians did wear hazmat suits (see image below), because they had the resources at their disposal to have them on hand. And in Season 4, the characters going to the Upside Down are driven by necessity and the intention of making their stay as brief as possible.
