Is the old Mac in Target Disk Mode?
If you connected the drive too late in the process (if using Setup Assistant or Migration Assistant), then you can go back a step or two, then move forward again so macOS scans for connected devices again.
You may need to manually mount the old hard drive using Disk Utility.
You can remove the old SATA drive and connect it to the new Mac using a USB to SATA Adapter, drive dock, or enclosure. If you have a new M1 Mac, then some USB accessories are not compatible with the new M1 Mac due to the USB controllers contained within the accessories. The JMicron USB chipset is one that is known to have compatibilty issues with an M1 Mac, but there may be other specific models of other brands of USB chipsets that are also affected. Unfortunately I haven't really seen any good article written about this issue. I know that the older Tripplite USB to SATA Adapter works on an M1 Mac.
It also assumes the old drive is still healthy and not beginning to fail.
Make sure to have frequent and regular backups. This is even more important when using an SSD since it is impossible to recover accidentally deleted data from an SSD once the Trash has been emptied plus an SSD can fail at any time without any warning signs. If the new Mac ever has a hardware issue, then only an expensive professional data recovery service will have any chance of possibly recovering the data from the non-functional Mac.