Other than a urinary tract infection what else can cause a constant slight sense of urgency to urinate?

Many things: There are a number of causes; a partial list includes, stones, neurological diseases, inflammation of adjacent gastrointestinal or gynecological organs, fluid collections or blood clots in the pelvis, some diseases causing bladder lining inflammation called interstitial cystitis, or even bladder tumors can do this. Remember infections are not just bacterial, fungal or viral infections also can.