My answer is based on your example:
For instance, if the Jammat for the 4 Furz of Zhur is at 1:30pm, and the calling of prayer is 1:16pm
If the call of the prayer starts at 1:16 you should (this recommended based on the hadith, which recommends saying what the muadhin says and in this hadith teaches us how) wait until it ends, and perform the dua' quoted in this hadith (here a version with a transliteration). Then you can pray an optional prayer. Assuming you mean the 4 rak'a sunnah before dohr you should of course perform it before the call for Iqamah for the mandatory prayer (so this is the timezone were you can perform it) and after the adhan.
This means if the time for the prayer has started earlier than the call for adhan -for example according the prayer chart dohr is at 1:13 and in your mosque they make the call for adhan at 1:30 and Iqamah at 1:45- you are allowed to perform it before the adhan but after 1:13, but i won't recommend it because it might create confusion and you also have enough time after the adhan in your mosque.
This is how you should do it if you already were in the mosque when the adhan call was performed.
If not it is recommended to perform the two rak'a greeting of the mosque first than the optional sunnah for Zohr (see also my answer on Do i read the sunnah rakkats in the mosque for details about this).
Also note that any sunnah prayer which is related to a fard prayer has a specific timing (this means it can only be performed before or after the prayer, for example for Zohr there is a sunnah qabilya (before) and a sunnah ba'diyah (after), but for 'Asr there's only a sunnah qabilyah and the time always starts with the adhan or entering of the time for the mandatory prayer). On the other hand if you are in a Muslim country you should have in mind that praying these kind of sunnah prayers at home is recommended (see my answers on Will my salah be accepted if I pray at home? and Is it obligatory to pray Taraweeh in the mosque?).
See also Sunnah prayers; how many rak'āt?
, Where is praying 2 and 2 sunnah prayers before Zuhr comes from? and Offer all sunnah before farz.