I'm 6'2, so it must be 6'2 night.
Everything depends by what you mean by a close grip.18" 12", 8", or 6" is the usual gripping space. But a Smith Machine is not the best piece of equipment for close gripping BP's (or even regular BP's). It does not follow the natural "S" curve when benching and does not allow enough natural flexability in the hands/wrist and forearms. Not to mention the shoulder girdle. Too much unneeded stress leads to a injury. Most close grip BP's are started from around the upper ab/lower chest region.
As HedgeHog suggested, if you have a regular straight bar, than you may want to use that instead. If still a problem, than if your a gym member they should have a ticep bar (Olympic style is longer and let's more weight to be used) that allows the fingers to face you when pressing off a bench (Hammer grip). Much more of a natural grip for BP'ing. Can always use a pair of DB's in the finger facing you position (Hammer grip), which may be better than a regular straight bar in the long run. Gives a more nature twist and flow to the exercise. Less stress on all the joints.Touch the DB's together at the top of the lift. Of course alternate with regular BP's and/or closer grip dips to hit the pec's hard. Good Luck.