Is it inappropriate to ask a professor for multiple letters of recommendation (at different times)?
It is not a problem in principle to ask for letters of recommendation at different periods of time. However, one should certainly make an effort to keep the recommender aware of career developments since the letter was originally written. (However, the relevant time scale over which old letters of recommendation become "outdated" is much longer than one year!)

For the most part, in the digital age, we keep letters of recommendation "on file" and recycle and refresh the content as warranted, so it's not a herculean effort to write a "new" letter after a while.