Even though an away congregation is supporting an evangelist full time because the one he is preaching for is small, shouldn't they (the congregation he is working with) still try to provide some support for him in whatever way?
If another congregation is supporting a preacher fully, then the preacher doesn't need additional funds. While in Corinth, Paul received support from other congregations and did not ask for support from Corinth. "Did I commit sin in humbling myself that you might be exalted, because I preached the gospel of God to you free of charge? I robbed other churches, taking wages from them to minister to you. And when I was present with you, and in need, I was a burden to no one, for what I lacked the brethren who came from Macedonia supplied. And in everything I kept myself from being burdensome to you, and so I will keep myself" (II Corinthians 11:7-9). Paul argued in I Corinthians 9 that he had the right to ask for pay from Corinth, but he chose not to do so. Therefore, it is between the preacher and those supporting his work who pays for what.
Typically, as a congregation grows, it offers to take over portions of the preacher's support, relieving the other congregation of its burden. Hopefully, it eventually gets to the point that it can not only support its own preacher full time but also repay the favor shown it by supporting preachers in other areas that cannot afford a preacher.