The UC system is set up so nobody needs an attorney. Even if you had one, he or she has no means to bend the law to make quitting to take care of an ill pet (for example) a "necessitous and compelling reason" under UC statutes.
Would additional evidence be admissible? No. Appealing the decision made following the hearing is an administrative appeal only. In other words, the State will review the testimony and evidence provided at the hearing and determine if an appropriate decision was made under the relevant statutes.
Is the 2nd appeal usually a waste of time and money? This will cost you nothing in money and time other than the cost of the postage to send a letter requesting the appeal. Overturning a decision made following a hearing can happen but it's pretty rare. The hearing adjudicators know what they're doing.
If you'd care to share the reason why you resigned, it would be a lot easier to tell you whether it's worth the stamp to appeal the decision.
the relaxed, non-formal atmosphere of the appeal process misled me to believe that the referee would base his ruling on common sense Judges - evening UC hearing adjudicators - make rulings based on the law, not common sense. You may have had a very compelling personal reason to quit your job but it appears it wasn't for a reason that UC statutes would allow you to collect benefits or that your former employer should pay for those.
A person, who is nice to you, but rude to a waiter, is not a nice person. (This is very important. Pay attention. It never fails.)