This is usually due to the wear on the rear sprockets. The tooth profile and pitch no longer matches the new chain, causing it to ride up on the sprocket teeth, worst on the smaller high gear sprockets which wear the most.
Some cassettes allow the smaller sprockets to be changed, but if not, it usually means a new cassette of sprockets to match the new chain. The larger front chain wheel tends to be less affected.
What mileage have you done?