Like I alluded to in my comment where I said it doesn't have to be mercenary: if you would otherwise have a reason to get together with somebody in person, make plans to do it. It can be one-on-one or with others, but the point is that the main reason for the meeting is social, i.e. you're doing exactly what you would have normally done. Then, during the customary "what's new with you?" part of the conversation you can let them know what type of work you do and that you're looking for. The main point is that you're actively seeking to get together with your friends, rather than waiting for them to call you. This is just keeping your network warm and planting seeds; you're not grilling them for job opportunities, and there is nothing inauthentic about it since it truly is a social call. Think of it as a slow burn, or the long game, or whatever metaphor works.

When it comes to people you don't know well enough to ask to get together with in person, shoot them an email or text. There's nothing wrong with being up front with what you're looking for, e.g. "Hey Joe, this is Mike (friend of your acquaintance Jane from Company XYZ), just letting you know I'm focusing on technical project management [or whatever] these days, wondering if you know of any part time opportunities in the field?" Receiving this request from an acquaintance is light years more acceptable than receiving it from a total stranger.