If he's not on the 40 man roster, "last possible second" is too late.
But, yes, you can call a player up after the regular season and before playoffs start. You can also send him back down if/when you're eliminated.
If he's good, I use 25/55 as a starter. PC of 55/65. He'll give you 4 innings, never get a win but will have a fine WHIP/ERA.
Hardball Dynasty – Fantasy Baseball Sim Games - Player Profile: Will Mahoney Sucks that he's 23-52 for his career but I'll take 120 innings of 1.13/3.15 all day. And I want to make sure I get as many of those innings as I can. That can only be accomplished by starting.