We report on the search for a CPT- and Lorentz-invariance-violating coupling of the He3 and Xe129 nuclear spins (each largely determined by a valence neutron) to posited background tensor fields that permeate the Universe. Our experimental approach is to measure the free precession of nuclear spin polarized He3 and Xe129 atoms in a homogeneous magnetic guiding field of about 400 nT using LTC SQUIDs as low-noise magnetic flux detectors. As the laboratory reference frame rotates with respect to distant stars, we look for a sidereal modulation of the Larmor frequencies of the colocated spin samples. As a result we obtain an upper limit on the equatorial component of the background field interacting with the spin of the bound neutron b(⊥)(n)<8.4 × 10(-34) GeV (68% C.L.). Our result improves our previous limit (data measured in 2009) by a factor of 30 and the world's best limit by a factor of 4.