Hola, saludos de Felipe
[url=https://flic.kr/p/2mMqGK1][img]https://live.staticflickr.com/65535/51709889966_211efe991e_c.jpg[/img][/url][url=https://flic.kr/p/2mMqGK1]BOSQUES EN INVIERNO[/url] by [url=https://www.flickr.com/photos/invierno123/]freme_3[/url], en Flickr
What is your 'default lens'?
I suggest you try the Sony 55-210mm OSS: it's fairly cheap but has all the bells and whistles (AF, stabilisation, OK image quality). I doubt it'll be sufficient for subjects 1-2km away though: will be good only for identifying subjects. And then rises the question: do you need AF? If yes, your budget needs to be expanded considerably compared to the 55-210 and I doubt these options qualify as a 'cheap zoom len' to you. If no, maybe @XKAES can help you find a decent option.