PNG is already using the DEFLATE algorithm (same that ZIP uses). You can use tools like pngout or pngcrush which try to find better compression, but make sure to test that your game toolkit can still open the resulting PNG files. You can reduce the bit depth from 24 bits to 8 (eg with Photoshop or opt-png in littleutils), but this will reduce the colors to 256 (like GIF). Other than that, PNG is already a compressed format and further compression won't help much. You can scale the texture atlas down and possibly endure a quality loss at runtime. Otherwise, using smaller images or images with less details will result in smaller PNGs.
You can more easily reduce the export data size. First step is to use a binary export. Second step is to compress the binary export (deflate, lzma, lzo, etc). After that, see the user guide for optimization.
Metrics - Spine User Guide: Performance