Adding outlet to existing outlet controlled by switch
Connection
Plug 1 currently has 3 wires:

White neutral back to the panel
White neutral to Plug 2
Black switched hot to the switch

Run a cable (black/white) from Plug 1 to New Plug.
Add a white pigtail (short wire) and connect one end to both existing neutral wires and to the white wire going to New Plug. Connect the other end to one of the existing neutral screws (doesn't matter which one, except see GFCI issue below) on Plug 1.
Add a black pigtail and connect one end to the existing switched hot and to the black wire going to New Plug. Connect the other end to the existing hot screw on Plug 1.
New Plug gets black on the hot screw and white on the neutral screw. The usual color-coding on receptacles is Black/Hot to Brass and White/Neutral to Silver.
GFCI
All new outdoor receptacles require GFCI. If you already have GFCI &quot;upstream&quot; (either as part of the circuit breaker or at an inside receptacle that is prior to the first outdoor receptacle) then you don't have to do anything special.
If you do not already have GFCI then you need to add it. The catch is that with your existing wiring scheme, you need to add it either at the source (somewhere inside) or at both Plug 1 and Plug 2. New Plug gets protected by Plug 1, provided it is wired correctly, which can get complicated so ask a new question if needed.
Note also that GFCI/receptacle combinations placed outside are likely to not last as long as GFCI/receptacle combinations (or GFCI/circuit breaker) placed inside, due to weather.
Speaking of weather, all outside receptacles should be weather resistant and also include an in-use cover.