Cvetković, Rajković, and Ivković proved that the Hankel transform of the sequence of sums of adjacent Catalan numbers is the bisection of the sequence of Fibonacci numbers. Here, we find recurrence relations for the Hankel transform of more general linear combinations of Catalan numbers, involving up to four adjacent Catalan numbers, with arbitrary coefficients. Using these, we make certain conjectures about the recurrence relations satisfied by the Hankel transform of more extended linear combinations.