You have several options, but most of them involve some form of interpolation.
Option 1 is to use short-term interest rate minus realized year-over-year inflation. Since inflation is published monthly with a one-month lag, you'd need to do some (very harmless) interpolations.
Since interest rate embeds forward expectations, a potentially better option is to use expected inflation (e.g., consensus economic forecasts). Depending on the data source, these could be of slightly higher frequency and are generally more timely. But in the end, some interpolations will still be needed.
The chart below shows two estimates, one using realized inflation and one with expected inflation:
I would NOT recommend using short-term real yield from TIPS, since these linkers are very illiquid and very technical in nature (with their yields reflecting a confluence of seasonality, illiquidity premium, and other idiosyncratic factors). The chart below shows 3-month TIPS yields over time: