While measuring inflation appears very complex, waiting a month over and over to see what is happening feels like a sub-optimal way to get data to be used to try to control inflation. I'm sure the FED doesn't just seek data on the next to last day of the month, but instead is getting new data all the time. Why could they not be updating the inflation picture in real time, as they get data? The way they are working now is much like trying to drive a car while given a peek out the windshield every five minutes.
As a corollary thought, were the FED to have real time data on inflation, they could micromanage rates rather than waiting a month and then having to jump rates in an attempt to catch up to inflation. If inflation jumps a tiny bit Tuesday then lift rates a tiny bit immediately. I assume that there would be some good reasons not to move rates frequently, but if the big picture intention is to limit inflation to a desirable zone, agility would be a valuable tool. It feels like we are spending 29 out of every 30 days driving out of control.
Yes, more timely inflation measures do exist. For example, academics have used new data-gathering techniques, referred to as "big data", to create daily inflation measures; see, e.g., the Billion Prices Project (https://www.hbs.edu/faculty/Pages/item.aspx?num=52242). However, there are trade-offs between using these measures and the official (monthly) price indexes, which are the CPI (produced by the Bureau of Labor Statistics) and PCE inflation (produced by the Bureau of Economic Analysis). While the big-data measures are more frequent, the official measures have broader coverage of services and goods that are not sold on-line.
Whether the Federal Open Market Committee (FOMC) would benefit from making interest rate decisions on a daily basis is a separate question. The interest rates that matter most for consumers and businesses are arguably not short-term rates (e.g., like the overnight Federal Funds rate), but longer-term rates like those for mortgages or commercial debt. These longer-term rates are influenced by the (expected path for the) Fed's policy rate, which allows the Fed to use monetary policy to affect financial conditions. Notably, these longer-term rates can adjust at a high frequency (much more frequently than daily) and incorporate, among other things, market participants' views of how the Fed will respond to incoming data. Even though interest rate decision are only made twice a quarter in the US, communication by policymakers between meetings can (and does) influence longer-term rates.