will there be a new trailer for us to look forward to?

Yeah for sure. There will more than likely be a third trailer. My guess is that the third trailer will get released in April or very early May.

It feels like the marketing for the American Godzilla is rolling, so a new trailer is possible.

It would not surprise me at all if they did release a new trailer before release of the film itself. Usually with films of this size and nature it's common to have at least 3 trailers. One as the "Teaser" the second for plot or tone of the film. The third for a "Sizzle reel" to kinda get the anticipation really geared up before the movie opens. But of course I could be wrong.

Don't get your hopes up. If there is another trailer it will be the week right before the movie. It's called reminding people about the movie after several months. Again don't get your hopes up. There will most likely be no trailer until May.

I think the teaser spots will be ramped up pretty high by the week before the movie. As for a final trailer I'm trying to remember when the third trailer to these movies typically come out. Most likely within a month or exactly a month before the release. Most big blockbusters get two trailers, but if a studio feels it could benifit from a fourth one that's possible as well.