You must ensure your removalist is properly qualified to offer you insurance. Since the passing of the Financial Services Reform Act (FSRA), it is illegal for any non-qualified person to give you financial advice or sell you a financial product.
Your qualified removalist will be able to offer you a Product Disclosure Statement (PDS) and a Financial Services Guide (FSG) and answer any questions you have concerning the insurance of your goods whilst in transit, that means between houses.
If a removalist is unable to do this, they may not be qualified to sell you insurance.
Independent insurance is always best and often cheaper