The contract for the making of the improvement may be let either before or after the making up of the equalization of the assessment roll, and warrants or bonds may be issued against the local improvement district fund either before or after the equalization of the roll as in the judgment of the council or commission may best subserve the public interest.

If after the assessment roll is made up and equalized, based in whole or in part upon an estimate of the cost of the improvement, and it is found that the estimate was too high, the excess shall be rebated pro rata to the property owners on the assessment roll, the rebates to be deducted from the last installment, or installments, when the assessment is upon the installment plan.

If it is found that the estimated cost was too low and that the actual bona fide cost of the improvement is greater than the estimate, the city council or commission after due notice and a hearing, as in case of the original equalization of the roll, may add the required additional amount to the assessment roll to be apportioned among the several parcels of property upon the same rules and principles as if it had been originally included except that the additional amount shall be added to the last installment of an assessment if assessments are payable upon the installment plan. The same notice shall be required for adding to the assessment roll in this manner as is required for the original equalization of the roll, and the property owner shall have the right of appeal.