Migrating Oracle smallfile to bigfile tablespace
You have a few options.

Create your new bigfile tablespaces, then either:

1) Move the tables one by one with:

alter table mytable move tablespace bigfile_tablespace;


Remember to move indexes too!

alter index myindex rebuild tablespace bigfile_index_tablespace;


2) Export the database with Data Pump, drop the existing objects, then reimport using a remap_tablespace clause, for example:

impdp remap_tablespace=OLDSMALLTS1:NEWBIGTS1,OLDSMALLTS2:NEWBIGTS2 directory=mydir dumpfile=mydumpfile.dmp logfile=mylogfile.log


3) Use DBMS_REDEFINITION to move the objects. An example of this can be found in my answer to another question, here.

Once done, drop all of the old objects and then the tablespaces.

If you can afford the downtime, datapump will be the easiest option.