REST API Call "PUT" how to Update packet filter rule with IP address and not to replace existing

Overwriting the existing rule is expected because the rule is represented as a single string (irrespective of a number of conditions joined by OR). This applies also to the equivalent tmsh command (modify net packet-filter <rule> ".....rule....").

To modify the rule, you need to GET the rule, compose a new rule from the current configuration, and PUT it.

I would use jq to create a rule with an additional "src host" (assuming that the rule consists of just "src host xx.xx.xx.xx") like this (SatPktRule is the name of the rule):

# Informatioal. Check the current rule.
$ curl -sku $PASS https://$HOST/mgmt/tm/net/packet-filter/SatPktRule | jq '.rule'
"(src host 10.10.10.10 or src host 10.10.10.20)"
 
# Add "src host 10.10.10.30" to the current
$ curl -sku $PASS https://$HOST/mgmt/tm/net/packet-filter/SatPktRule | jq '.rule | rtrimstr(")") + " or src host 10.10.10.30)"'
"(src host 10.10.10.10 or src host 10.10.10.20 or src host 10.10.10.30)"

P.S. If you do not need to change the action and order properties, you can use the PATCH command instead. You only need to specify the rule.