Doubts about replacing an LCD display
HD44780/ST7066 LCDs of a specific geometry are usually compatible (especially the very common 16x2), but watch out for the orientation and the V0 voltage. (If your circuit doesn't have a contrast adjustment you could add a 10k trim potentiometer.)

So yes, you can most probably make the connection with wires. Or search some more, I've seen 16x2 LCDs with pins on both the top and the bottom side.