It was once kind of rare, but now it isn’t uncommon for Catholic parishes to have men’s groups or Bible study groups or prayer groups and the like that meet outside of Mass time. The nice thing is that most parishes now have web pages that make it easy to see what sorts of groups are available and when they meet. Also, many people who want a “small group” Mass seek out daily Masses, which are typically offered early in the morning (6 am to 9 am, depending on the parish), noon or maybe once or twice a week in the evening (6 pm-7 pm region). College campuses also have daily Masses that are open to the public, but you’re less likely to make friends there if you’re not a student than if you go to a daily Mass at your own parish.
The other sure way to make new acquaintances who might become friends, of course, is to volunteer, especially in corporal works of mercy done as a group. I went to Mass at a parish last week at which the pastor pointed out that you’re only half way there if all you do is pray and come to Mass. You need to do works of mercy, too.