One common misconception (not saying you are making it) about inflation is that it means rising prices. In fact, it is a theoretical construct about the rise of the general price level = a decrease in the value of money. All we have are proxies (CPI) for this construct and it is often really hard to know if a change in the proxy actually means a change in the general price level or just a change of relative prices as a result of eg. supply shocks.
So sure, a month where CPI starts to increase faster could mean that inflation is speeding up, but it could also just be a result of changes to the real economy that influence the prices of things in the CPI basket. In that sense, something that might look like an increase in the price level might end up being just an adjustment of relative prices (that might readjust after supply issues have been fixed).
