We have performed Co59 NMR measurements of CeCoIn5 down to ultralow temperatures. We find that the temperature dependence of the spin-echo intensity provides a good measure of the sample temperature, enabling us to determine a pulse condition not heating up the sample by the NMR pulses down to ultralow temperatures. From the longitudinal relaxation time (T1) measurements at 5 T applied along the c axis, a pronounced peak in 1/T1T is observed at 20 mK, implying an appearance of magnetic order as suggested by the recent quantum oscillation measurements [H. Shishido, Phys. Rev. Lett. 120, 177201 (2018)10.1103/PhysRevLett.120.177201]. On the other hand, the NMR spectrum shows no change below 20 mK. Moreover, the peak in 1/T1T disappears at 6 and 8 T in contrast to the results of the quantum oscillation. We discuss that an antiferromagnetic state with a moment lying in the a-b plane can be a possible origin for the peak in 1/T1T at 5 T.