If I play Hostile Takeover and my opponent still has units on that battlefield, can I move more units to that battlefield before the showdown starts?

Can't move in and play Hostile Takeover during showdown as it is slow speed.

No. Once Hostile Takeover resolves while your opponent still has units on that battlefield, the Combat Showdown is staged and opens immediately during the cleanup after the spell resolves — there is no window to Standard Move additional units in first. Standard Moves are only legal in a Main Phase Open State, never during a Showdown or Combat (Rule 144.1.c), and the cleanup that follows the control change opens the staged combat with no intervening action window (Rule 323.13). This matches the ruling for the same family of effects: after Showstopper moves one unit to attack, "the showdown must be resolved first before you can move additional units" (FAQ #1026). One things you can do: bring units in during the showdown itself via abilities that specifically allow moves/plays mid-showdown (e.g., an effect that plays or moves a unit there, which joins the combat as an attacker per Rule 464.2.c.3.a).