I've had two excellent versions of good ol' calamari friti in recent years:
1) In several places in Connecticut, they serve calamari with sliced banana pepper rings. The heat and vinegar are exactly what are needed to offset the greasiness of calamari. Definitely a good approach (the oiliness of giardinera would probably not be a good substitute)
2) At Eccola, in Parsippany, NJ, they served a calamari chinense (sp?), which had a sweet, garlic and ginger sauce poured over it. Almost like a General Tso kind of sauce. Outstanding.