I was thinking about this earlier and I have a question about inflation. Could increasing the interest rate cause inflation to rise a bit in the near term? My

I was thinking about this earlier and I have a question about inflation. Could increasing the interest rate cause inflation to rise a bit in the near term? My reasoning for this is that given that banks were borrowing at near zero, could they have had no real reason to put all the borrowed money to work since it wasn't costing them anything to hold it in reserve for later when the rates did increase? Now that the rates are increasing would they not have to use the money a bit to ensure they stay ahead of the interest rates. I was also thinking that there is a threshold at which banks wouldn't have any more money that is just sitting there and having to borrow at higher rates reduces their demand for new funds from the fed thus undoing this initial effect to the hike. Hopefully this isn't completely naive. Please let me know if I'm misunderstanding how the fed and banks relationship works.