If you could survive a very close pass with a black hole you could certainly achieve speeds that would produce a noticeable relativistic effect.
Hypervelocity stars seem to be the product of a double boost from a lost companion stars and a supermassive black hole. Serious speeds like 1500 miles per second.
http://en.wikipedia.org/wiki/Hypervelocity_star#Hypervelocity_stars
Holy mackerel, that’s from New York city to Los Angeles in less than 2 seconds. One one thousand - Two one thous - we’re here. Freakin’ insane speed.
I looked up the distance to the moon....238,900 miles. So 1,500 mps you would get there in 159.6 second or 2.64 minutes lol lol About the length of a standard pop song. That’s incredible.