  Physical properties of an ultracold Fermi gas in the temperature-coupling
phase diagram can be characterized by the contact intensity C, which enters the
pair-correlation function at short distances and describes how the two-body
problem merges into its surrounding. We show that the local order established
by pairing fluctuations about the critical temperature Tc of the superfluid
transition considerably enhances the contact C in a temperature range where
pseudogap phenomena are maximal. Our ab initio results for C in a trap compare
well with recently available experimental data over a wide coupling range. An
analysis is also provided for the effects of trap averaging on C.
