Folklore has no scientific basis.* The "recent and well-known conference report" does. You've taken that further by adding details and formally proving results. Congratulations, you've advanced science.
In parallel, "[t]he main result was also proved by someone using a different approach." You've been scooped. Some of your work's novelty has been lost.
Nonetheless, given that the published work appears in "a good journal," we can infer that the folklore (in this instance) was worthy of formally establishing in science. Whether an alternative approach is similarly worthy, we can't know for sure, only opinions can be offered. I suggest asking the journal's editor whether they'd consider publishing your alternative approach. E.g.,
Dear Prof X,
To my dismay, I discovered my work on ABC has been scooped by Prof Y et al. in their article (entitled "DEF"), published in volume N issue I of Journal Name, over which you preside.
Perhaps all isn't lost: Prof Y et al. use approach blah, whereas I use approach blah-blah. I wonder whether you'd consider publishing my work as an alternative approach (subject to peer-review, of course)?
If not, could you perhaps suggest an alternative venue?
Many thanks for your support,
* Details matter. Yet, folklore (and, more generally, story telling) omits, looses, or never possessed details. Folklore has no scientific basis. If folklore sufficed, we'd have no need for science. "Nevertheless," Paul Taylor notes, "if some younger person is so presumptuous as to write out a proper proof and attempt to publish it, they will get shot down in flames." A theorem, by definition, demands proof, without such it is but an idea, a conjecture at best. "The ossification of a caste system – in which one group has the general ideas and vision while another toils to realize that vision – is no way for a subject to flourish," remarks Clark Barwick. Maybe I'm naive, perhaps ivory towers unassailable. That'll tarnish science. Building upon complaisant folklore will lead to failure, there's no rigor.