Reports suggest that Emperor Maximilian has been supplying Confederate Texas via blockade-running shipments originating from the port of Veracruz. The government's first recourse against this is to simply demand that the shipments stop, trusting that the threat of American intervention on its own would be enough to intimidate the Mexican royalists. If that failed, the President would have the option to send a detachment of the Union Navy to enforce a tighter blockade of the port, with the objective of stopping the shipments from the source or, failing that, presenting a show of force to make the royalists back down.
This latter course would effectively be taking sides in the ongoing civil war in Mexico. It would earn the friendship of the republican forces, which would be highly useful for the Texan campaign, as they control the actual border with Texas and would certainly be willing to assist if America took such decisive action against their enemies. On the other hand, Veracruz is also the primary port through which the Mexican Empire receives supplies from its French patrons, so blockading it risks conflict not just with the Empire but with France as well.
Overall, there are a wide range of possible outcomes to pursuing the Mexican connection. The best-case scenario is that Maximilian simply accedes to American demands, taking enough pressure off of the Texas front that it and Reconstruction can proceed simultaneously. The worst-case scenario is a French counter-demand to back down or face war. It is up to the President and his advisors whether the risk is worth the reward.