You can pause a process with SIGSTOP and later continue it with SIGCONT. (See
kill -l for a list of signals and use
ps to get the process ID).
However there is a fair chance that this will not work. E.g. if you have a router between your source and destination which notices and 'cleans' a dead connection. Most SOHO routers seem to do this because cleaning connections means less memory use. (And thus they can build cheaper routers).
If that happens you will either need to copy again with scp or you can use rsync.
Rsync has the advantage that it compares already existing files and does not needlessly copy data. In your case it should detect an already present, partially uploaded file and only upload the remainder of the file
Syntax would be
rsync -v -e ssh /home/my_file firstname.lastname@example.org:~
-v for verbose. -e "ssh options" to specify the ssh as remote shell.
If your data is compressable then you might want to add
-z to compress the data.