I think just about everywhere I've worked has a cost of living raise, even entry level at restaurants. In any case this is somewhat of a contraindication to the economic planning undertaken by the Federal Reserve and their aims with inflation, which is to increase the monetary base to allow expansion. What you're proposing would, I believe in theory, close that gap as it costs employers more without necessarily increasing revenue whereas, with the current modality, things remain closer to baseline.
However, if instead of diffusing the cash through the government, it was directly granted to citizens by the government/FR one might would presume similar results as the current modality, just more beneficial to the individual. However this takes on some degree of complication: there are manifold ways to disburse it, for instance we could make a straight cut for every individual, weigh it based on income, weigh it based on wealth, and one could justify weighing in either direction - presumably the wealthy are better arbiters of economy than are the needy... But also the needy are needy and it may help alleviate some degree of disparity.
Also, inflation is difficult to measure. The CPI is not a great mark. Expanding the monetary base by 2% on paper may not result in a 2% increase in CPI. CPI is, I reckon, used as a check on the dollar's relative value, which can change dynamically as a result of cultural, economic, and political shifts on top of monetary policy. So resultsay vary, but considering market behavior over the pandemic, I think it points to a rapid dissemination of whatever money is given. I think that would probably, actually be a really efficient means to deal with inflation, however a lot of that money is actually doled out in subsidies, grants, et cetera. So a new system of public goods funding would necessarily have to be set up.