The prevailing thought is that even if you relined it with Spanish cedar, the cedar of which the chest is made will permeate everything. So you'll have mild Spanish cedar next to the cigars, all being overpowered by the red (or whatever) cedar from the chest. The only way you could likely do it, according to prevailing thought, would be to completely line the interior with a barrier (plastic or the like) that would completely barricade the red cedar smell from the interior. Then line that barrier with Spanish cedar. Naturally that's a lot of work, and not guaranteed to work.
I've never spoken to anyone who has tried to retrofit a red cedar chest, and I think it could be done with enough work. But the work required likely would make it no longer worth it unless you really, really, really wanted that as a piece of furniture.