Let us denote by COILS(υ) a (3,2,1)-conjugate orthogonal idempotent Latin square of order υ, and by ICOILS(υ, n) an incomplete COILS(υ) missing a sub-COILS( n). A necessary condition for the existence of an ICOILS(υ, n) is υ ⩾ 3 n + 1. An ICOILS(υ, 1) is equivalent to a COILS(υ), and the necessary condition for its existence has recently been shown by the authors to be sufficient for all υ ⩾ 4 with the exception of υ = 6 and the possible exception of υ = 12. Two of the above authors have previously shown that for n>1, an ICOILS(υ, n) exists if υ = 3 n + 1 or υ ⩾ 8 n + 42. Moreover, it was also shown that, for 2 ⩽ n ⩽ 6, an ICOILS( v,n) exists for all υ ⩾ 3 n + 1 with some possible exceptions. The main purpose of this paper is two-fold. First of all, for 2 ⩽ n ⩽ 6, we substantially reduce the number of possible exceptions and show that, in particular, the necessary condition is sufficient for n = 4,5 and 6 except possibly when (υ, n) = (30, 5). Secondly, we show that for n ⩾ 1, an ICOILS(υ, n) exists for all υ ⩾ ( 13 4 )n + 88 , which gives a general bound much closer to the necessary condition.