Actually, the hospital/doctor is NOT what second aid originally is. And first responder aid is also not what first aid is about.

First aid is what is meant as the actual first aid that can be provided by anyone with minimal training. As the concept came from the military the "first aid" was meant to be applied by fellow soldiers - this is were the (rather surgical trauma oriented average first aid kit got most of it's content of). (Unit-level embedded medics came far later) In a civilan setting nowadays it includes the basic life support provided by non-specialist first responders and depending on the definition it might include BLS ambulances (which are not seen as professionals sometimes)

Second aid was what a medical professional does - a paramedic, corpsman or doctor and can also include a casualty collection post or similar structures. Definitions here split up a bit - depending on which authors you read second aid (aka the second aid stage) does not include any intrahospital care but does prolonged care scenarios(cases you cannot evacuate to a medical facility for longer times,like 24h+x). Other authors do include basic hospital care or similar facilities that provide some but not all treatment.

After that tertiary or definitive care takes over - that's when shit definitely gets fixed if it's fixable. That will always include intrahospital care.

To get back to your question what would be in a second care kit: Things needed for prolonged care usually. I did work in a prolonged care scenario for a while and can give you an overview what we had in our "prolonged care" kits: Urinary catheters (people need to pee even with a broken spine and we need to monitor urine production. And you can fix almost everything with it...you fix holes in hearts with them, can use them as a feeding tube,etc.), more and more permanent pain medication, suturing kits, surgical drainages,lots and lots of desinfectant, more iv fluids, heating solutions both for the patient and the fluids, medication for more permanent care (strong antibiotics, heparin to avoid pulmonary embolism from immobilisation), Military medics often include patient to patient blood transfusions kits, but I worked on the civilian side. These would easily enable one to care for patients for up to 72h hours. (Luckily 36h is the most I had to spend with a critically ill patient).

Source: Consultant paramedic.