Connected neighborhoods in products

Let X and Y be metric continua. We consider the following property (*): if M is a subcontinuum of X×Y such that πX(M)=X and πY(M)=Y, where πX and πY are the respective projections on X and Y, then M has small connected neighborhoods in X×Y. Property (*) has been studied by D. P. Bellamy, J. M. Łysko and the first named author. In this paper we continue studying property (*) in products of continua. We prove: (a) the product of homogeneous continua having the fixed point property has property (*); (b) the product of a solenoid and any Knaster continuum has property (*); (c) there exists a Kelley continuum X such that X×[0,1] does not have property (*); and (d) the product of a chainable Kelley continuum and [0,1] has property (*).