  The physics of the strongly interacting Hubbard chain (with $t/U \ll 1$) at
finite temperatures undergoes a crossover to a spin incoherent regime when the
temperature is very small relative to the Fermi energy, but larger than the
characteristic spin energy scale. This crossover can be understood by means of
Ogata and Shiba's factorized wave function, where charge and spin are totally
decoupled, and assuming that the charge remains in the ground state, while the
spin is thermally excited and at an effective "spin temperature". We use the
time-dependent density matrix renormalization group method (tDMRG) to calculate
the dynamical contributions of the spin, to reconstruct the single-particle
spectral function of the electrons. The crossover is characterized by a
redistribution of spectral weight both in frequency and momentum, with an
apparent shift by $k_F$ of the minimum of the dispersion.
