It is difficult to predict if this is necessary.

If you have good stability and a homogeneous protein this may be enough. If you observe aggregates, dimeric forms or if your laboratory analyses require high purity, it is advisable to do this second step Size Exclusion Chromatography.

You can deposit purification fractions on SDS-PAGE gel after your affinity chromatography to validate if it is necessary or not.

His-tag chromatography is not as specific as vendors want you to believe, there is always non-specific binding, and binding of native proteins with enough histidine residues. If the obtained purity is not enough after IMAC, then a second chromatography is regularly followed. SEC is expensive and always pursued only in laboratories. IEX is regularly used as second chromatography.

As stated above, firstly with his-tag, a lot of non-specific proteins will also bind and co-elute (unless the expression of your protein is very high such that the relative amount of impurities becomes very less)- this can be checked on sds-page. Secondly, after dialysis it could be possible that the protein aggregates or forms higher mol wt species- this will depend upon the protein and the specific buffer condition used- this can be checked with native gel.

Gel filtration can be planned accordingly.

But it will also depend upon the use of this purified protein. eg., if you want to use it for some assay where impurities do not really matter, you can skip SEC.

Usually, after the initial capture step with your first column during purification, you'll need an additional polishing step. His-tag chromatography generally works well, but even after thorough washing, non-specific binding can remain, necessitating a further purification step.

You have several options available. Personally, I often choose size-exclusion chromatography if the impurity bands are more than 20 kDa away from the target protein. The choice between more expensive or less costly methods largely depends on your sample volume. For example, a 16/600 Superdex 75 or 200 column may not be as expensive as you expect.

Alternatively, you might consider hydrophobic interaction chromatography or another ion exchange column. Ultimately, the best choice depends on your sample conditions, so it's hard to determine definitively without additional details.

Please let me know if you'd like to discuss this further in more detail—I'm happy to help.