Savings rates are tied to interest rates, not inflation. But they won't necessarily move up immediately when the Fed starts raising rates next week, though. They will move up when there is a need for banks to attract more money, in order to make loans. Americans saved a lot during the pandemic, and that extra input to banks is sitting around waiting to be loaned out, or withdrawn and spent.
https://www.statista.com/statistics/246268/personal-savings-rate-in-the-united-states-by-month/A great alternative right now is I Bonds, which are paying 7.12%. The downsides are: you can only buy $10,000 per calendar year, (Plus up to $5,000 Through a federal tax refund) you have to hold them a minimum of one year, and if you redeem in less than 5 years, you give up 3 month's interest.
The interest rate paid resets every 6 months, but It's looking like, if anything, it may go up a bit next time, in May.
Another option: SoFi, which just recently became a "real" bank, is offering 1% on up to $50,000, if you establish a direct deposit into the account.