Inflation does not affect (relatively speaking) the poorest more. On the contrary, it's a tax on people that have a lot of cash or cash-equivalent assets. That's not the poor. The poor have either very little cash or even negative (via debt), so inflation benefits them.
And you can see this historically, too:
Historical inflation rate chart: http://www.aboutinflation.com/_/rsrc/1369736825466/inflation...
Wealth gap inequality: https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...
You can literally see how the wealth gap inequality nose dives in the high-inflation environment of the 70s and then continually increases after low-inflation rates become the norm for decades after around 1982.
Inflation is a tax, should be seen as such but is a tax on the rich in cash (I keep insisting on the term cash because that's what inflation taxes, if you are rich in other assets inflation may or may not result in an effective taxation of your wealth). Monetary and fiscal policy should be designed with this in mind.