We study the online sorting problem, where n real numbers arrive in an online fashion, and the algorithm must immediately place each number into an array of size \((1+\varepsilon ) n\) before seeing the next number. After all n numbers are placed into the array, the cost is defined as the sum over the absolute differences of all \(n-1\) pairs of adjacent numbers in the array, ignoring empty array cells. Aamand, Abrahamsen, Beretta, and Kleist introduced the problem and obtained a deterministic algorithm with cost \(2^{O\left( \sqrt{\log n \cdot \log \log n +\log \varepsilon ^{-1}}\right) }\) , and a lower bound of \(\varOmega (\log n / \log \log n)\) for deterministic algorithms. We obtain a deterministic4 algorithm with quasi-polylogarithmic cost \(\left( \varepsilon ^{-1}\log n\right) ^{O\left( \log \log n\right) }\) . Concurrent and independent work by Azar, Panigrahi, and Vardi achieves polylogarithmic cost \(O(\varepsilon ^{-1}\log ^2 n)\) .