Less than a month before the end of the 2012 election, the first-term Kentucky senator is embarking on his most vigorous effort yet to expand his national profile.

Over the past week, Paul used his political committee — RAND PAC — to launch television ads in the Ohio, West Virginia and Florida Senate races, hammering incumbent Democrats on foreign aid. That offensive will intensify in the coming weeks, Paul advisers said, with an additional two or three Senate races on the Republican’s target list.