I do sometimes search [using DuckDuckGo or Google] for products that I'm not sure who might sell, and then end up at Amazon. And that path does not get you into the "Smile" system.
But, as long as you notice, prior to the final checkout page, that you are not using 'Smile', its easy to fix. Even though you may already have items in your Amazon 'cart', its not necessary to blow that all away and start over. Simply replace the "www" part of the URL in your browser with "smile" and proceed to checkout. As long as the final checkout page has the 'Smile' logo, a portion of the transaction ends up in the WW&F bucket.
My primary browser is Firefox, and I find that it is quite easy to get to "smile" (when I think about it in advance) by just typing "smi" and then clicking the "smile" selection that Firefox offers up. Of course, that only works because Firefox remembers from my previous use. So once you start doing it, your browser may make it a bit easier.