The easiest step for you is to go for the '52 part (e.g. AT89S52), which has twice as much RAM - 256 bytes - although the upper 128 bytes is to be used indirectly only (a great place to place the stack into it).

If you find out you need more, there are models with "internal external" memory, built into the chip, but accessed using MOVX. Of these, the 'RD2 sub-family (e.g. AT89C51RD2 of Atmel, or P89V51RD2 of NXP) are pin-compatible with the "classical" '51, and have also more extra features (lots of FLASH, a complex timer section called PCA, twice the speed - 6-clock instruction cycle). There are a lots of other, non-pin-compatible, low-voltage etc., but that's added complexity which is maybe not necessary in your case.

Nevertheless, you have to find out yourself, which of these is available at your place. There's no point in suggesting you something what you can't reasonably buy.

JW

PS. If the 256 bytes of RAM of '52 is not enough, you might want also to consider attaching some memory - EEPROM or FRAM - serially, e.g. via I2C. Unfortunately, there are no easily available SRAMs with I2C.