This paper investigates the thermoremanent magnetization (TRM) of the Martian lithosphere assuming that the upper part of the lithosphere acquired TRM in the early history of the planet and in the presence of the core field (the primary magnetization), whereas the lower part has been gradually magnetized by the magnetic field of the upper part as it has cooled below the Curie temperature (secondary magnetization). The secondary magnetization is relatively weak if the magnetic properties of the lower lithosphere are taken to be similar to that of the extrusive basalt near the oceanic ridge axes on Earth. However, if the Martian lower lithosphere is more magnetic, e.g., by an order of magnitude, the secondary magnetization can be significant, although still weaker than the primary magnetization of the upper part. Such a highly magnetic lower lithosphere does not seem plausible by terrestrial standards, and it remains to be confirmed whether it is geologically plausible for Mars. I also examine a lithospheric model with a strongly magnetic layer beneath the crust, 25 times more magnetic than the extrusive basalt, representing a possible ilmenite-rich layer with a considerable amount of hematite-ilmenite lamellae. I assume that the layer was not magnetized by the core field because it was hotter than the Curie temperature of the lamellae when the core dynamo was active. It acquired secondary magnetization later in the presence of the magnetic field of the crust. Despite adopting an extraordinarily high magnetic layer which seems implausible by terrestrial standards, the secondary magnetization of the layer has minor contributions to the observed magnetic anomalies. In all models considered the primary magnetization is ~20--30 A/m and has the dominant contribution to the observed magnetic anomalies. It is worth noting that the radial components of the primary and secondary magnetization are parallel, whereas the tangential components are in opposite directions. Because the strong magnetic anomalies in the south likely arise from the tangential magnetization, considering the secondary magnetization requires even stronger primary magnetization.