After a beloved pet has passed away, the motivation and timing to invite another animal into your life is very personal.
There is no right or wrong way to consider this. Some people need the spirit of animal around them and feel a reduced quality of life without their presence. The capacity to love and deeply appreciate the profound contributions of sentient beings does not end with their death. It can leave an aching void in an empty heart that nothing else will satisfy.
It is not disloyal to the life of your departed pet to give shelter to another of his brethren, but a way to further honor his place and that of all animal in our lives. Nothing and no other creature will ever replace the experience of your beloved friend. But the heart does not permanently contract with loss.
Thankfully, there is no end to the capacity of the human heart which expands again with the glory of renewal.
Some people, feeling the agony of loss, think they could never go through this again and vow to never adopt another animal. But once discovered and celebrated in the soul, the capacity and need to love an animal does not disappear. There is always the future chance that some other sentient being will move you to offer a home. Decide to not decide during this difficult time when perspective is clouded by grief. When the time is right, the eyes of some sweet and knowing creature may capture you again.