No, you shouldn't be concerned. You do need to add another asset for the 2024 improvements. You cannot change the original cost basis because the depreciation wouldn't compute correctly, as the rental and improvements have different placed-in-service dates. So, by adding the new improvements, your cost basis will go up.
If the improvements are personal property (not real property), choose the option for carpet, furniture, etc. Under the half-year convention, your depreciation will be zero, and thus no recapture. The half-year convention means you get a half year of depreciation in the year of purchase, and a half year of depreciation in the year of disposition, thus no depreciation if bought and sold in the same year.
