This paper is concerned with verification methods for numerical solutions of linear systems. Many methods for the verification require switches of rounding modes defined by the IEEE 754 standard. However, the switches cannot be supported in several computational environments. In such cases, Ogita-Rump-Oishi's method can work on such environments. Recently, Rump developed new error estimates of floating-point summation and dot product. The aim of this paper is to improve Ogita-Rump-Oishi's error estimates by using the error estimates by Rump. In addition, the computational cost of our method is comparable to that of Ogita-Rump-Oishi's method.