An increase in supply would reduce wages, unless it also increases demand. If you think about wages in cities vs rural areas, you’ll see that most of the time more people = more economic activity = higher wages.
Where this breaks down, is if there’s barriers of entry that prevent immigrants from participating in the economy fully. If immigrants aren’t allowed to legally work or start business (as happens with some asylum seekers or ‘illegal’ immigrants) then they are forced to compete over a small pool of off-book / cash-in-hand jobs, which could see a reduction in wages without a significant increase in overall economic activity.