We divorced by khul'. If we want to get back together, can we do so without a new marriage contract?

No; khul' is an irrevocable (ba'in) divorce, so the two of you can only reunite through a new marriage contract (akad nikah) and a new mahr. Rujuk by words alone does not apply after khul', even if the iddah has not ended. The good news is that if the khul' was not the third talaq, you do not need to marry someone else first. A new akad requires a wali, two witnesses and registration like an ordinary marriage, so deal with the district Islamic religious affairs office. Use this chance to set a new understanding about what caused the earlier separation. Also make sure the remaining talaq count is worked out correctly so there is no confusion later. Read more on the types of talaq page.