I suggested those LCD character displays as something close to a direct replacement to your existing display. I very much prefer the OLED I'm using but I hear what you are saying about its size. There is a slightly taller OLED (128x128), which might work, especially with two or three side-by-side.
For larger displays, it seems like graphical LCD is the only option.
This 3.5" ezLCD-313 seems like it might be good if the refresh rate can be high enough. It would certainly be very easy to program, as the controller provides a serial interface. It does not have a bright backlight (80 nits) but that shouldn't be a factor when lit from the front, since it's transflective. I'm tempted to try it out since it looks so easy to use.
This 2.2" LCD might work and is smaller, but has a backlight of 200 nits, which is not enough to show up in sunlight (from what I read, that requires about 800+ nits, like these).
I think this Sharp 3.7" LCD (model LS037V7DW05) might be the very best. It is transflective and seems to offer their best indoor/outdoor performance and includes a resistive touchscreen. Also available without a touchscreen (model LS037V7DW06), which also has brighter backlighting. But it would require some proper development to pair a controller to the raw display. It generally has been my experience that drivers are available for most of the commonly used controller chips; that's how I am using my OLED. But I don't know anything about a controller appropriate for the Sharp displays.
Edit: It looks like 4D Systems has some interesting options including a slightly larger, 160x128 OLED with a controller on it. I wish I had started with that instead. They have a few different LCDs but none immediately jumps out as great in the daylight.
