Disowning the process and redirecting STDOUT from an already backgrounded process isn't feasible for most users. This SO answer covers the topic pretty well.
Disassociating the process from the current shell isn't so hard: you're looking for
disown. From the manpage:
The shell exits by default upon receipt of a SIGHUP. Before exiting,
an interactive shell resends the SIGHUP to all jobs, running or
stopped. Stopped jobs are sent SIGCONT to ensure that they receive the
SIGHUP. To prevent the shell from sending the signal to a particular
job, it should be removed from the jobs table with the disown builtin
(see SHELL BUILTIN COMMANDS below) or marked to not receive SIGHUP
using disown -h.
SHELL BUILTIN COMMANDS
disown [-ar] [-h] [jobspec ...]
Without options, each jobspec is removed from the table of
active jobs. If jobspec is not present, and neither -a nor -r
is supplied, the shell's notion of the current job is used. If
the -h option is given, each jobspec is not removed from the ta‐
ble, but is marked so that SIGHUP is not sent to the job if the
shell receives a SIGHUP. If no jobspec is present, and neither
the -a nor the -r option is supplied, the current job is used.
If no jobspec is supplied, the -a option means to remove or mark
all jobs; the -r option without a jobspec argument restricts
operation to running jobs. The return value is 0 unless a job‐
spec does not specify a valid job.
If you want to learn more about what
disown actually do, you can check out this question. (disclaimer: shameless self-promotion)