Typically, you will have excess oligo, and you can consider any method that selectively captures the conjugate but not the free oligo I.e. anything that selectively binds to the antibody. What will be the easiest depends to some extent on the antigen. If you can put antigen on beads, you can do immunoaffinity purification. If not, protein A or protein G might be worth considering. With these methods you have to expose the bound conjugate to acidic elution buffer, which may be okay but needs to be validated. Otherwise you need standard chromatography equipment and a medium that can differentiate Ab-oligo from oligo. It is hard to make a recommendation here as the type of Ab and its surface properties, and the size of the oligo and degree of conjugation all need to be considered. Probably protein G is the easiest route in most cases, but some conjugates may not bind because of steric hindrance by the conjugated oligos. Pre-binding of Ab to antigen columns and then labelling with oligo is perhaps the most sophisticated approach.

On small scale, size exclusion chromatography using hydrophilic sorbents with the exclusion limit about 50kD should work well. The conjugate should get eluted in the void volume, while unbound excess oligo should be retained.

Thanks for your suggestions