Normally, with three categories, you will obtain an intercept, reflecting the log odds of the outcome in the reference category, and two effect terms, indicating how the log odds for the other two categories differ from the reference.
You can change the contrasts used in the model to change how this information is encoded (see https://stats.oarc.ucla.edu/r/library/r-library-contrast-coding-systems-for-categorical-variables/ ).
However, what I think you're looking for is just an estimate of the log odds for every category, or in other words, an intercept for each category. This is done by suppressing the overall intercept, using the syntax
outcome ~ -1 + predictor in R.
As mentioned in the comments, it is technically possible to fit a model where with an intercept and effects for each category. However, this model is not identified, since there is no unique best-fitting set of parameters. This model can be estimated, though, using regularisation - applying a small penalty to the weights (but not the intercept) such that parameters with smaller squared magnitudes are preferred (this can also be achieved using something called the Moore-Penrose pseudoinverse; see here). This yields a unique solution: the intercept is the average of the log odds across categories, while the effects for each category indicate how much each category differs from the average. It should be easy to see why this solution has the smallest squared parameter sizes (again, not counting the intercept).
Again, though, this is an advanced topic, and unless you really know what you're doing, you probably don't want this! I only know of this approach because some people use it to analyse fMRI data using the popular but idiosyncratic SPM software.