Not really. Lee didn't tell his employer where he lived. He was living on Beckley under an assumed name. Not even his wife knew where he was living. Even if he thought they could have suspected him as early as 12:34 PM, why would he think they would know about the Beckley room 20 minutes later?
You'd have to ask Lee that.
Still, no harm in having the cab go past the rooming house before getting out.