In 1908 Hermann Minkowski improved Special Relativity Theorie by replacing the Lorentz transformation by his own differential formula. Einstein followed him because now the speed of the proper observer could vary. Later Einstein made use of Minkowski’s formula in his General Relativity.
Another advantage was that the Ehrenfest paradox was solved. The Lorentz length contraction would impossibly deform a turning rigid cylinder.
More basically, in Minkowski’s formula, the proper observer (in a smal moving reference frame) travels in a wider Euclidian reference frame. Now the clock- and the twin-paradoxes disappear: both frames are not equal anymore, as Einstein decided.