Unfortunately, the speed at which characters enter the screen for battles is already as speedhacked as possible. I took a look into the time period between when you stop moving for a random battle and when the battle map appears in view, and I couldn't find anything useful. I did get one loop hacked, but I really don't know if it's going to help out at all. In the meantime, I did speedhack the fade-in and fade-out sequence when entering or exiting the menu, so that should be faster (if it wasn't already at full speed).
Here are two patches you can try. The first is the same as the other one with just the menu speedhack added. The second includes the one hack that may or may not speed up the "buzz" thing you're talking about. Let me know if it helps at all (though I don't think it will).
C0FA0464|Final Fantasy 3 (V1.1) (U)|0|0|0|0|0|0|28236=42DC,B7A3=EAEA,B80A=EAEA,568 =42FC,11A28=42DC,31372=42DC
C0FA0464|Final Fantasy 3 (V1.1) (U)|0|0|0|0|0|0|28236=42DC,B7A3=EAEA,B80A=EAEA,568 =42FC,11A28=42DC,31372=42DC,13DA1=42D3
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