I've only basic training in chemistry, since my competences are in Electrical Engineering, but I'd like to tackle your question from another angle.
I'm assuming your question is sort of an X-Y problem and you are not really interested in increasing the efficiency of an actual lemon-based cell.
From an engineering POV, what you want from a power source is, simply speaking, to extract the maximum amount of energy without losing much in the process.
There are 3 basic quantities involved in this: energy, power and efficiency. Voltage and/or current are of much lesser importance, since there are electronic methods that can boost one of those at the expense of the other.
For AC generators, a transformer can increase/decrease the voltage while decreasing/increasing current (with a given load) at the same time.
For DC generators such as batteries, you can add a switching DC/DC converter that can step the voltage up or down as needed with high efficiency.
So the point is: is yours an efficient method to improve either the power or the energy output of your cell?
As already pointed-out, your "multi-cell" arrangement can be modeled as a series connection of cells with every cell having a resistor in parallel. This latter models the fact that the electrolyte of the various cells is shared among them.
Those parallel resistances represent an additional load to the cells. This wastes power and energy. Are these losses negligible or not? To answer this you should consider other losses in the system, e.g. the internal resistance of the electrolyte, the contact resistance of the connection wires and the resistance of the wires themselves.
To do a meaningful comparison, you have to know the value of those parallel resistances, and this is probably tricky business.
But remember, the energy stored in a cell depends on the volume of the electrolyte (well, probably on its mass, but assuming a more or less constant density...), so you should ask if there is a better method to suck out the energy out of a given volume of electrolyte. Well this is what is already done when you need more energy: you build a bigger cell. And if you need also more power, you increase its current rating (since it's voltage can't be changed) by increasing the surface of the electrodes.
If you need higher voltage, for a given power, you simply put a DC/DC boost converter between the cell and the load. Nowadays, this is what is done in lots of products that can be powered with a single AA or AAA cell, for example.
In this way you avoid entirely those extra losses represented by those parallel resistances.
As already pointed out in other answers and comments, your idea could work, but it's inefficient. To increase the efficiency you should increase those parallel resistances and this means more distance between pairs of electrodes. This implies a need for a bigger volume of electrolyte.
This is also (probably) inefficient because there are already other well-tested means to extract the energy from the same amount of electrolyte and provide to the end-user (the load) the voltage level it needs to work properly.
Moreover your system is much more complicated to set-up, even if you could make the losses negligible, since the output voltage can only be set by multiples of the cell voltage, and you would need a DC/DC converter (or an DC/AC converter if your load needs AC) anyway to produce the voltage the load needs.