You should consult an oral surgeon and get that piece of the tooth removed immediately..

Your pain might be because of d dry socket. As there was infection hope you took a proper course of a good antibiotic. Please visit your dentist ASAP. The dentist will do d needful either remove the remaining piece if its big. Normally a small tip if left behind will not cause much problem as it will get resorbed by itself over a period of time

I know your doctor told you to come back after 1 month... That was assuming everything is fine.. Did you get back to him again stating your current status of excruciating pain ? If not do so immediately.. I am sure he will have a solution as he did the initial extraction..

Dear somtimes a very small piece of tooth which we call it as root stays inside the socket accidentally or knowingly which usually may cause secondary infections and disturbs normal wound healing also and somtimes if the very small piece of root stays inside and u the consultant dentist does a good dressing and let u go it won't create a problem as the root piece automatically resorbs gradually, bt if the affected tooth which u r talking was decayed or infected and it ws almost hlf of the previous tooth thn u must go and extract the tooth within few days as u r getting severe discomfort at tht region as it might b a dry socket,

U go to other doctor and extract that tooth. Dont leave that portion of tooth

If there is any broken tooth piece left, it should be removed. Get an X-ray done and consult a good oral surgeon