This paper proposes constructing a confidence set for the date of a structural change at the end of a sample in a mean shift model. While the break fraction, the ratio of the number of observations before the break to the sample size, is typically assumed to take a value in the (0,1) open interval, we consider the case where a permissible break date is included in a fixed number of observations at the end of the sample and thus the break fraction approaches one as the sample size goes to infinity. We propose inverting the test for the break date to construct a confidence set, while critical values are obtained by using the subsampling method. By using Monte Carlo simulations, we show that the confidence set proposed in this paper can control the coverage rate in finite samples well, while the average length of the confidence set is comparable to existing methods based on asymptotic theory with a fixed break fraction in the (0,1) interval.