In this context, an 'array' means any arrangement of antenna elements that are combined at RF. They don't have to be aligned. You're not the first to use multiple elements aimed in different directions to provide better coverage.
Combining antenna elements at RF tends to make the pattern more complicated (peaks and nulls). You want an isotropic pattern which is as simple a pattern as can be imagined (a simple sphere) - but that can be very difficult to achieve in practice. The isotropic antenna is often called 'the mythical isotropic antenna' (perhaps a bit harsh).
If you're using one ferrite core for your antenna, then it should only have two major nulls - one off each end (just like any old AM Broadcast Band receiver). Therefore, you should only require (at most) one more ferrite core plus receiver to provide complete coverage (you shouldn't need three).
If you're dealing with an on/off keyed signal, then the baseband is basically DC. That's the easiest possible baseband. Depending on the on/off key rate, maybe you could have two of your ferrite cores arranged orthogonally, and then use a switched receiver back and forth between the two antennas. This would allow your system to detect signals from any direction without having to pay for two complete receiver strips. RF switching at 13MHz isn't rocket science.
Another option would be to abandon the linear ferrite core and investigate other options. Active antennas work fairly well at HF frequencies. They just need a short probe and a high impedance RF amplifier. The probe might still have nulls.
Maybe a ferrite that is not linear? I would guess that a curve ferrite might be somewhat null-free (a guess...).
What is your application (to the extent that you're permitted to talk about it)? If I recall correctly, 13.56MHz is an 'Industrial' frequency and is often used for induction heating. Are you trying to measure field strengths ? If so, then just buy a suitable, calibrated probe...