Was there anything different past beating The Legend of Zelda twice

Commonly with virtual console games, you can hit the center button (Wii menu?), and exit the game. This will automatically save the state of the game. Next time the game loads you'll begin at that spot; however, after reloading once the save state is no longer available -- resetting the system won't allow you to resume from that state again.

It is highly likely that the game's initial design had some fancy thing in mind, like dumping the user into a different room than the main entrance. However, that was infeasible due to limitations in the code used for the game's map format.

Also, each room had only one destination, even if there were multiple "entrances". By "entrance", I refer to the idea of not only staircases (found in the overworld and the underworld), but also caves. The overworld was not substantially different than the underworld in this respect.

I have a bit more information about this topic, thanks to a friend who dedicated a summer to reviewing such tidbits in great detail, and later gave me a summary of some of his findings.

Many players may have realized that many of the rooms look like they have the same basic layout for the center (ignoring certain details, like the walls/doorways, monsters, and so forth.) These similarities are no accident. Since there was substantial economic incentive to try hard to minimize required memory space in this day when each bit was much more expensive than today, the game used multiple techniques to minimize the amount of data required for each room. Each room had a single reference to a pre-defined layout. Each layout was made up of columns that were re-usable in other layouts. And even each column was made up of tiles that could be repeated: the first bit of a tile could be used to specify whether it was simply a repeat of the previous tile (just North of the one just used). So even though CPUs were far slow than today, the data that made up the floor centers had at least 3 references to look up (room layout number, column pattern number, tile type), which all resulted in re-use of bits.

The code was designed so that if Link ever walked off of the bottom of the Underworld Map, then he would go back to the overworld. Every dungeon entrance is on the very edge (and, specifically, on the very bottom) of the map, and that's why.

If the format of map data was altered so that more bits were used to support multiple destinations, Nintendo might not have felt that the available memory could fit the five different sets of 16x8 collections of screens/rooms that they did end up using. (Two of those were for the First Quest underworld, one for the Overworld which mostly re-used the same memory for both quests, and then two more collections were used for the Second Quest underworld.) If Nintendo had stuck to their vision of allowing multiple destinations, we might not have been so fortunate as to be able to enjoy the "Second Quest".