Assuming that, after buying the options, you will no longer trade in the underlying or options, it only makes sense to buy a put, in addition to the call, if you want to bet that the price of the underlying (PU) will experience a big increase or a big decrease. Note that you will be paying the premiums of both the call and of the put. Therefore, the increase or decrease in the PU needs to be high enough to generate a payoff greater than the sum of both premiums.
Of course you can choose the strikes and the volume of the call and the put to have a delta neutral position at the time you buy the two options. But this only means that your bet roughly equally to an increase or a decrease in the PU. However, as soon as the UP changes, your delta will not be zero: this is not a delta neutral strategy, i.e., a strategy of continuously trading to keep the delta equal to zero during the whole life of the options.
In a delta-neutral strategy during the whole live of the option(s) you are in theory completely hedged. Therefore, you will not make or lose any money (as long as the observed volatility is equal to the implied vol during the life of the option), which means that you are not betting on an increase of the PU.
The moral of the story: You cannot bet and hedge at the same time. Actually, hedging is a strategy that aims to avoid any bet, and therefore, any risk.
If you want to bet that the PU will increase and not decrease, you have only to buy and to pay the premium of the call.