FLC OF AN MLG ACR ACFT OVERSHOT ASSIGNED CLB ALT DUE TO MISSETTING THE ALT ALERTER.

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CLRED FOR TKOF FROM HOU; RWY 22R; TURN TO HDG 030 DEGS. (ALT CLRED TO WITH CLRNC WAS 4000 FT.) DURING TKOF BRIEF BY ME; I VERIFIED 4000 FT LEVEL OFF. AFTER TKOF WE BEGAN OUR CLB AND R TURN TO 030 DEGS. IN OUR CLB WE WERE GIVEN TFC; AN LTT AT 2 O'CLOCK CLBING; HAS US IN SIGHT. I HAD THE LTT ON TCASII CLOSING FROM 3 O'CLOCK; 1000 PLUS BELOW; FO HAD A VISUAL ON THE LTT. TWR CLRED US TO DEP. AT THAT TIME WE WERE CLBING THROUGH 5500 FT. CHKING IN WITH DEP I RPTED CLBING TO 6000 FT THROUGH 5600 FT (6000 FT WAS SET IN THE WINDOW). DEP CTL ASKED WHO CLRED US TO 6000 FT. WE WERE CAUGHT; SAID NOTHING. DEP IMMEDIATELY CLRED US TO A HIGHER ALT. AFTER CHKING WITH THE TWR; NO ONE HAD CLRED US TO 6000 FT. I BELIEVE WE HAD THE WRONG ALT SET IN THE ALT WINDOW; EVEN THOUGH I LOOKED AT IT; AND RPTED 4000 FT DURING THE TKOF BRIEF. THERE WAS NO CONFLICT; AS RPTED BY DEP. WE CONTINUED WITHOUT INCIDENT TO OUR DEST. COPLT WAS FLYING THE ACFT. WE MAY HAVE CAUGHT THE ERROR HAD WE NOT BEEN CONCENTRATING ON THE LTT TFC.