If you're doing a workout routine like 5/3/1 or PPL (Push-Pull-Legs) with a home gym, you may need to pick your accessory exercises based on equipment you have. In a commercial or powerlifting gym, you often have a wide variety of machines and specialty equipment to choose from. If your equipment selection is limited, but you still need to add in some push accessory exercises, check out this list for some ideas.
No extra equipment is required for push-ups, which makes it perfect for an accessory exercise for any workout routine. I like to work sets of 20 push-ups as supersets with non-push compound movements like squats or deadlifts to get some additional push volume in my workout. Variations can be used to make push-ups easier (on your knees instead of your toes) or harder (wider or narrower hand positioning, or one-handed).
In terms of pushing in a direction away from your body, dips are one of the few movements that have your arms pushing downward. This makes it a great accessory exercise by making it work on a slightly different plane of movement from other push exercises. While having a dedicated dip station or rack-mounted dip handles helps make the movement more stable, you can also use two chairs with weights on them or something like a washing machine and a dryer. To make dips harder, you can wear a weighted vest or hold a dumbbell between your feet. You can also use gymnastic rings for added instability.
Lying triceps extensions are sometimes called "skullcrushers", as you're lowering weights down to your head before extending back up. Triceps extensions can use barbells, dumbbells, or an EZ-bar, depending on what you have available. These are most often done laying down, but if you are outside or have a high enough ceiling then you can do them standing as well for a slight variation.
Side lateral raises are great for building your shoulders. It's easiest to do these with dumbbells, but if your weight plates have handles then you can just hold the plates while raising your arms to the side. If you are trying to increase the weight but are struggling to not cheat the movement by moving more of your body, try leaning forward and placing your head on the wall or a pillar to isolate your shoulders. Lateral raises can be quite humbling, as each small increase in weight can really drop the number of reps you are able to do.
Since a lot of accessory movements lean towards isolation of specific muscles, one of the best ways to isolate the pecs is to do chest flyes. Like lateral raises, these are best done with dumbbells but can work with weight plates that have handles. I've found that keeping your elbows slightly bent at the same angle throughout the movement helps to reduce strain on the elbows. Chest flyes are often difficult to make progress in weight since each small jump is tough, but it can really help build pec strength.
The barbell overhead press is often used as one of the main movements in a workout routine, but variations on a shoulder press make for great accessory movements. If you typically use barbells for this, try dumbbells or vice versa. Using OHP as an accessory is also an opportunity to work in different rep ranges than usual, like doing 8-12 reps in a set for lighter weight than you normally would.
Like the shoulder press above, the bench press is often a main movement in workout routines. Using variations, like incline bench press or dumbbell bench press, can help hit the chest and triceps in slightly different ways. Other options, like floor press or board press, are also useful as home gym push accessory exercises.